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Properties

Label 4-10e2-1.1-c21e2-0-0
Degree 44
Conductor 100100
Sign 11
Analytic cond. 781.075781.075
Root an. cond. 5.286565.28656
Motivic weight 2121
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.04e3·2-s + 1.00e5·3-s + 3.14e6·4-s + 1.95e7·5-s − 2.05e8·6-s + 1.32e9·7-s − 4.29e9·8-s + 7.55e9·9-s − 4.00e10·10-s + 2.18e10·11-s + 3.15e11·12-s − 4.12e11·13-s − 2.72e12·14-s + 1.95e12·15-s + 5.49e12·16-s − 1.10e13·17-s − 1.54e13·18-s − 5.20e13·19-s + 6.14e13·20-s + 1.33e14·21-s − 4.47e13·22-s + 8.09e13·23-s − 4.30e14·24-s + 2.86e14·25-s + 8.43e14·26-s + 1.55e15·27-s + 4.18e15·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.980·3-s + 3/2·4-s + 0.894·5-s − 1.38·6-s + 1.77·7-s − 1.41·8-s + 0.721·9-s − 1.26·10-s + 0.254·11-s + 1.47·12-s − 0.829·13-s − 2.51·14-s + 0.877·15-s + 5/4·16-s − 1.33·17-s − 1.02·18-s − 1.94·19-s + 1.34·20-s + 1.74·21-s − 0.359·22-s + 0.407·23-s − 1.38·24-s + 3/5·25-s + 1.17·26-s + 1.45·27-s + 2.66·28-s + ⋯

Functional equation

Λ(s)=(100s/2 ΓC(s)2 L(s)=( Λ(22−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}
Λ(s)=(100s/2 ΓC(s+21/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22⋅522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 781.075781.075
Root analytic conductor: 5.286565.28656
Motivic weight: 2121
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 100, ( :21/2,21/2), 1)(4,\ 100,\ (\ :21/2, 21/2),\ 1)

Particular Values

L(11)L(11) ≈\approx 3.0657803233.065780323
L(12)L(\frac12) ≈\approx 3.0657803233.065780323
L(232)L(\frac{23}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1+p10T)2 ( 1 + p^{10} T )^{2}
5C1C_1 (1−p10T)2 ( 1 - p^{10} T )^{2}
good3D4D_{4} 1−33436pT+10326754p5T2−33436p22T3+p42T4 1 - 33436 p T + 10326754 p^{5} T^{2} - 33436 p^{22} T^{3} + p^{42} T^{4}
7D4D_{4} 1−189842188pT+29983126517490222p2T2−189842188p22T3+p42T4 1 - 189842188 p T + 29983126517490222 p^{2} T^{2} - 189842188 p^{22} T^{3} + p^{42} T^{4}
11D4D_{4} 1−21869194224T+ 1 - 21869194224 T + 12 ⁣⋯ ⁣6612\!\cdots\!66T2−21869194224p21T3+p42T4 T^{2} - 21869194224 p^{21} T^{3} + p^{42} T^{4}
13D4D_{4} 1+412060138772T+ 1 + 412060138772 T + 36 ⁣⋯ ⁣9436\!\cdots\!94pT2+412060138772p21T3+p42T4 p T^{2} + 412060138772 p^{21} T^{3} + p^{42} T^{4}
17D4D_{4} 1+11074763456364T+ 1 + 11074763456364 T + 79 ⁣⋯ ⁣7479\!\cdots\!74pT2+11074763456364p21T3+p42T4 p T^{2} + 11074763456364 p^{21} T^{3} + p^{42} T^{4}
19D4D_{4} 1+52072861011560T+ 1 + 52072861011560 T + 91 ⁣⋯ ⁣0291\!\cdots\!02pT2+52072861011560p21T3+p42T4 p T^{2} + 52072861011560 p^{21} T^{3} + p^{42} T^{4}
23D4D_{4} 1−80988875151948T+ 1 - 80988875151948 T + 37 ⁣⋯ ⁣2237\!\cdots\!22T2−80988875151948p21T3+p42T4 T^{2} - 80988875151948 p^{21} T^{3} + p^{42} T^{4}
29D4D_{4} 1−264315081456060T+ 1 - 264315081456060 T + 28 ⁣⋯ ⁣5828\!\cdots\!58T2−264315081456060p21T3+p42T4 T^{2} - 264315081456060 p^{21} T^{3} + p^{42} T^{4}
31D4D_{4} 1−2126016880188664T+ 1 - 2126016880188664 T + 36 ⁣⋯ ⁣8636\!\cdots\!86T2−2126016880188664p21T3+p42T4 T^{2} - 2126016880188664 p^{21} T^{3} + p^{42} T^{4}
37D4D_{4} 1+9721711577644724T+ 1 + 9721711577644724 T + 16 ⁣⋯ ⁣1816\!\cdots\!18T2+9721711577644724p21T3+p42T4 T^{2} + 9721711577644724 p^{21} T^{3} + p^{42} T^{4}
41D4D_{4} 1−221161031209870884T+ 1 - 221161031209870884 T + 25 ⁣⋯ ⁣4625\!\cdots\!46T2−221161031209870884p21T3+p42T4 T^{2} - 221161031209870884 p^{21} T^{3} + p^{42} T^{4}
43D4D_{4} 1+43823174002198412T+ 1 + 43823174002198412 T + 40 ⁣⋯ ⁣2240\!\cdots\!22T2+43823174002198412p21T3+p42T4 T^{2} + 43823174002198412 p^{21} T^{3} + p^{42} T^{4}
47D4D_{4} 1−4384129302297468pT+ 1 - 4384129302297468 p T + 22 ⁣⋯ ⁣9822\!\cdots\!98T2−4384129302297468p22T3+p42T4 T^{2} - 4384129302297468 p^{22} T^{3} + p^{42} T^{4}
53D4D_{4} 1+612435371018710692T+ 1 + 612435371018710692 T + 25 ⁣⋯ ⁣2225\!\cdots\!22T2+612435371018710692p21T3+p42T4 T^{2} + 612435371018710692 p^{21} T^{3} + p^{42} T^{4}
59D4D_{4} 1−4752208294289856120T+ 1 - 4752208294289856120 T + 41 ⁣⋯ ⁣0241\!\cdots\!02pT2−4752208294289856120p21T3+p42T4 p T^{2} - 4752208294289856120 p^{21} T^{3} + p^{42} T^{4}
61D4D_{4} 1−4748119038295687324T+ 1 - 4748119038295687324 T + 61 ⁣⋯ ⁣6661\!\cdots\!66T2−4748119038295687324p21T3+p42T4 T^{2} - 4748119038295687324 p^{21} T^{3} + p^{42} T^{4}
67D4D_{4} 1−22301925713082765436T+ 1 - 22301925713082765436 T + 56 ⁣⋯ ⁣5856\!\cdots\!58T2−22301925713082765436p21T3+p42T4 T^{2} - 22301925713082765436 p^{21} T^{3} + p^{42} T^{4}
71D4D_{4} 1−2201035283486261544T+ 1 - 2201035283486261544 T + 49 ⁣⋯ ⁣2649\!\cdots\!26T2−2201035283486261544p21T3+p42T4 T^{2} - 2201035283486261544 p^{21} T^{3} + p^{42} T^{4}
73D4D_{4} 1−56149494460474652548T+ 1 - 56149494460474652548 T + 34 ⁣⋯ ⁣2234\!\cdots\!22T2−56149494460474652548p21T3+p42T4 T^{2} - 56149494460474652548 p^{21} T^{3} + p^{42} T^{4}
79D4D_{4} 1+42382706176352134640T+ 1 + 42382706176352134640 T + 93 ⁣⋯ ⁣5893\!\cdots\!58T2+42382706176352134640p21T3+p42T4 T^{2} + 42382706176352134640 p^{21} T^{3} + p^{42} T^{4}
83D4D_{4} 1−1335343926003173268T− 1 - 1335343926003173268 T - 64 ⁣⋯ ⁣7864\!\cdots\!78T2−1335343926003173268p21T3+p42T4 T^{2} - 1335343926003173268 p^{21} T^{3} + p^{42} T^{4}
89D4D_{4} 1− 1 - 25 ⁣⋯ ⁣8025\!\cdots\!80T+ T + 18 ⁣⋯ ⁣7818\!\cdots\!78T2− T^{2} - 25 ⁣⋯ ⁣8025\!\cdots\!80p21T3+p42T4 p^{21} T^{3} + p^{42} T^{4}
97D4D_{4} 1+ 1 + 15 ⁣⋯ ⁣0415\!\cdots\!04T+ T + 16 ⁣⋯ ⁣9816\!\cdots\!98T2+ T^{2} + 15 ⁣⋯ ⁣0415\!\cdots\!04p21T3+p42T4 p^{21} T^{3} + p^{42} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−16.06818046620703130116479767172, −15.21912155159924401248132221141, −14.49413679433034969440174975376, −14.35683781409278282128308605557, −13.11934151912874543148122010925, −12.35154341522367395879062644715, −11.10058017522273584326129807660, −10.83923921608156252125173020419, −9.876867116057108418201930643452, −9.124673078026105705924920854388, −8.489474929124001929838925423219, −8.101501770434156580374706528036, −7.06369318748437245070050587819, −6.38797998370977605118697145051, −5.00712838776311479702800748208, −4.21010937299549460776127580880, −2.38643130931047642659572170273, −2.35774451691357030838514304202, −1.52849586582870041598465895635, −0.67386207780214672162600015047, 0.67386207780214672162600015047, 1.52849586582870041598465895635, 2.35774451691357030838514304202, 2.38643130931047642659572170273, 4.21010937299549460776127580880, 5.00712838776311479702800748208, 6.38797998370977605118697145051, 7.06369318748437245070050587819, 8.101501770434156580374706528036, 8.489474929124001929838925423219, 9.124673078026105705924920854388, 9.876867116057108418201930643452, 10.83923921608156252125173020419, 11.10058017522273584326129807660, 12.35154341522367395879062644715, 13.11934151912874543148122010925, 14.35683781409278282128308605557, 14.49413679433034969440174975376, 15.21912155159924401248132221141, 16.06818046620703130116479767172

Graph of the ZZ-function along the critical line