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Properties

Label 4-10e2-1.1-c21e2-0-2
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 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.04e3·2-s − 1.26e5·3-s + 3.14e6·4-s − 1.95e7·5-s + 2.59e8·6-s − 2.92e8·7-s − 4.29e9·8-s + 3.49e9·9-s + 4.00e10·10-s − 4.13e10·11-s − 3.98e11·12-s + 1.88e12·13-s + 5.99e11·14-s + 2.47e12·15-s + 5.49e12·16-s − 3.10e12·17-s − 7.16e12·18-s + 7.17e12·19-s − 6.14e13·20-s + 3.70e13·21-s + 8.46e13·22-s − 3.57e14·23-s + 5.44e14·24-s + 2.86e14·25-s − 3.86e15·26-s − 1.78e14·27-s − 9.20e14·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.23·3-s + 3/2·4-s − 0.894·5-s + 1.75·6-s − 0.391·7-s − 1.41·8-s + 0.334·9-s + 1.26·10-s − 0.480·11-s − 1.85·12-s + 3.79·13-s + 0.553·14-s + 1.10·15-s + 5/4·16-s − 0.372·17-s − 0.472·18-s + 0.268·19-s − 1.34·20-s + 0.484·21-s + 0.679·22-s − 1.79·23-s + 1.75·24-s + 3/5·25-s − 5.36·26-s − 0.166·27-s − 0.587·28-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: 22
Selberg data: (4, 100, ( :21/2,21/2), 1)(4,\ 100,\ (\ :21/2, 21/2),\ 1)

Particular Values

L(11)L(11) == 00
L(12)L(\frac12) == 00
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+126692T+464904286p3T2+126692p21T3+p42T4 1 + 126692 T + 464904286 p^{3} T^{2} + 126692 p^{21} T^{3} + p^{42} T^{4}
7D4D_{4} 1+41799812pT+9706833687364722p2T2+41799812p22T3+p42T4 1 + 41799812 p T + 9706833687364722 p^{2} T^{2} + 41799812 p^{22} T^{3} + p^{42} T^{4}
11D4D_{4} 1+41326831776T+ 1 + 41326831776 T + 36 ⁣⋯ ⁣0636\!\cdots\!06pT2+41326831776p21T3+p42T4 p T^{2} + 41326831776 p^{21} T^{3} + p^{42} T^{4}
13D4D_{4} 1−145157805556pT+ 1 - 145157805556 p T + 81 ⁣⋯ ⁣3881\!\cdots\!38p2T2−145157805556p22T3+p42T4 p^{2} T^{2} - 145157805556 p^{22} T^{3} + p^{42} T^{4}
17D4D_{4} 1+3100413932364T+ 1 + 3100413932364 T + 22 ⁣⋯ ⁣7422\!\cdots\!74pT2+3100413932364p21T3+p42T4 p T^{2} + 3100413932364 p^{21} T^{3} + p^{42} T^{4}
19D4D_{4} 1−377673402760pT+ 1 - 377673402760 p T + 36 ⁣⋯ ⁣5836\!\cdots\!58p2T2−377673402760p22T3+p42T4 p^{2} T^{2} - 377673402760 p^{22} T^{3} + p^{42} T^{4}
23D4D_{4} 1+357559990100052T+ 1 + 357559990100052 T + 87 ⁣⋯ ⁣2287\!\cdots\!22T2+357559990100052p21T3+p42T4 T^{2} + 357559990100052 p^{21} T^{3} + p^{42} T^{4}
29D4D_{4} 1+2454584202185940T+ 1 + 2454584202185940 T + 46 ⁣⋯ ⁣5846\!\cdots\!58T2+2454584202185940p21T3+p42T4 T^{2} + 2454584202185940 p^{21} T^{3} + p^{42} T^{4}
31D4D_{4} 1+4310677151243336T+ 1 + 4310677151243336 T + 16 ⁣⋯ ⁣8616\!\cdots\!86T2+4310677151243336p21T3+p42T4 T^{2} + 4310677151243336 p^{21} T^{3} + p^{42} T^{4}
37D4D_{4} 1+32389201650205724T+ 1 + 32389201650205724 T + 12 ⁣⋯ ⁣1812\!\cdots\!18T2+32389201650205724p21T3+p42T4 T^{2} + 32389201650205724 p^{21} T^{3} + p^{42} T^{4}
41D4D_{4} 1+16298035212712116T+ 1 + 16298035212712116 T + 71 ⁣⋯ ⁣4671\!\cdots\!46T2+16298035212712116p21T3+p42T4 T^{2} + 16298035212712116 p^{21} T^{3} + p^{42} T^{4}
43D4D_{4} 1+161845702253479412T+ 1 + 161845702253479412 T + 46 ⁣⋯ ⁣2246\!\cdots\!22T2+161845702253479412p21T3+p42T4 T^{2} + 161845702253479412 p^{21} T^{3} + p^{42} T^{4}
47D4D_{4} 1−153492993930726996T+ 1 - 153492993930726996 T + 18 ⁣⋯ ⁣9818\!\cdots\!98T2−153492993930726996p21T3+p42T4 T^{2} - 153492993930726996 p^{21} T^{3} + p^{42} T^{4}
53D4D_{4} 1−604198357317820308T+ 1 - 604198357317820308 T + 20 ⁣⋯ ⁣2220\!\cdots\!22T2−604198357317820308p21T3+p42T4 T^{2} - 604198357317820308 p^{21} T^{3} + p^{42} T^{4}
59D4D_{4} 1−3110197357974672120T+ 1 - 3110197357974672120 T + 32 ⁣⋯ ⁣1832\!\cdots\!18T2−3110197357974672120p21T3+p42T4 T^{2} - 3110197357974672120 p^{21} T^{3} + p^{42} T^{4}
61D4D_{4} 1+791142209760451676T+ 1 + 791142209760451676 T + 37 ⁣⋯ ⁣6637\!\cdots\!66T2+791142209760451676p21T3+p42T4 T^{2} + 791142209760451676 p^{21} T^{3} + p^{42} T^{4}
67D4D_{4} 1+5111819523383561564T+ 1 + 5111819523383561564 T + 33 ⁣⋯ ⁣5833\!\cdots\!58T2+5111819523383561564p21T3+p42T4 T^{2} + 5111819523383561564 p^{21} T^{3} + p^{42} T^{4}
71D4D_{4} 1+82388741313238482456T+ 1 + 82388741313238482456 T + 30 ⁣⋯ ⁣2630\!\cdots\!26T2+82388741313238482456p21T3+p42T4 T^{2} + 82388741313238482456 p^{21} T^{3} + p^{42} T^{4}
73D4D_{4} 1−10657918464992348548T+ 1 - 10657918464992348548 T + 27 ⁣⋯ ⁣2227\!\cdots\!22T2−10657918464992348548p21T3+p42T4 T^{2} - 10657918464992348548 p^{21} T^{3} + p^{42} T^{4}
79D4D_{4} 1−28314166125451369360T+ 1 - 28314166125451369360 T + 14 ⁣⋯ ⁣5814\!\cdots\!58T2−28314166125451369360p21T3+p42T4 T^{2} - 28314166125451369360 p^{21} T^{3} + p^{42} T^{4}
83D4D_{4} 1+125635249468995804pT+ 1 + 125635249468995804 p T + 38 ⁣⋯ ⁣2238\!\cdots\!22T2+125635249468995804p22T3+p42T4 T^{2} + 125635249468995804 p^{22} T^{3} + p^{42} T^{4}
89D4D_{4} 1− 1 - 47 ⁣⋯ ⁣8047\!\cdots\!80T+ T + 23 ⁣⋯ ⁣7823\!\cdots\!78T2− T^{2} - 47 ⁣⋯ ⁣8047\!\cdots\!80p21T3+p42T4 p^{21} T^{3} + p^{42} T^{4}
97D4D_{4} 1− 1 - 54 ⁣⋯ ⁣9654\!\cdots\!96T+ T + 92 ⁣⋯ ⁣9892\!\cdots\!98T2− T^{2} - 54 ⁣⋯ ⁣9654\!\cdots\!96p21T3+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

−15.85839538305185702401666177148, −15.12359892667593229874454707570, −13.64200455617642625052055268830, −12.99776444958657970857434160675, −11.66285262007658734522626524496, −11.61865778298513851339169783129, −10.73804741737209801406969588700, −10.48634307925554845713591773938, −9.118563010103317364979725732703, −8.492003153746732251829754573077, −7.88643975889491217582953307268, −6.79751258135651795703609932208, −6.04213711772108342139157952173, −5.64939506760527435706315109686, −3.86191063965913166125107408071, −3.42967016289538218632564970545, −1.76894170409024154823913311715, −1.11642112105821876100684601202, 0, 0, 1.11642112105821876100684601202, 1.76894170409024154823913311715, 3.42967016289538218632564970545, 3.86191063965913166125107408071, 5.64939506760527435706315109686, 6.04213711772108342139157952173, 6.79751258135651795703609932208, 7.88643975889491217582953307268, 8.492003153746732251829754573077, 9.118563010103317364979725732703, 10.48634307925554845713591773938, 10.73804741737209801406969588700, 11.61865778298513851339169783129, 11.66285262007658734522626524496, 12.99776444958657970857434160675, 13.64200455617642625052055268830, 15.12359892667593229874454707570, 15.85839538305185702401666177148

Graph of the ZZ-function along the critical line