Properties

Label 4-10e2-1.1-c17e2-0-1
Degree 44
Conductor 100100
Sign 11
Analytic cond. 335.703335.703
Root an. cond. 4.280444.28044
Motivic weight 1717
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 512·2-s + 1.76e4·3-s + 1.96e5·4-s − 7.81e5·5-s + 9.02e6·6-s + 2.76e7·7-s + 6.71e7·8-s + 1.44e8·9-s − 4.00e8·10-s + 6.45e7·11-s + 3.46e9·12-s + 2.89e9·13-s + 1.41e10·14-s − 1.37e10·15-s + 2.14e10·16-s + 1.58e9·17-s + 7.38e10·18-s + 4.42e10·19-s − 1.53e11·20-s + 4.88e11·21-s + 3.30e10·22-s + 4.87e11·23-s + 1.18e12·24-s + 4.57e11·25-s + 1.48e12·26-s + 1.88e12·27-s + 5.44e12·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.55·3-s + 3/2·4-s − 0.894·5-s + 2.19·6-s + 1.81·7-s + 1.41·8-s + 1.11·9-s − 1.26·10-s + 0.0907·11-s + 2.32·12-s + 0.984·13-s + 2.56·14-s − 1.38·15-s + 5/4·16-s + 0.0549·17-s + 1.57·18-s + 0.597·19-s − 1.34·20-s + 2.81·21-s + 0.128·22-s + 1.29·23-s + 2.19·24-s + 3/5·25-s + 1.39·26-s + 1.28·27-s + 2.72·28-s + ⋯

Functional equation

Λ(s)=(100s/2ΓC(s)2L(s)=(Λ(18s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}
Λ(s)=(100s/2ΓC(s+17/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 335.703335.703
Root analytic conductor: 4.280444.28044
Motivic weight: 1717
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 100, ( :17/2,17/2), 1)(4,\ 100,\ (\ :17/2, 17/2),\ 1)

Particular Values

L(9)L(9) \approx 14.7028178614.70281786
L(12)L(\frac12) \approx 14.7028178614.70281786
L(192)L(\frac{19}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1p8T)2 ( 1 - p^{8} T )^{2}
5C1C_1 (1+p8T)2 ( 1 + p^{8} T )^{2}
good3D4D_{4} 15876pT+685454p5T25876p18T3+p34T4 1 - 5876 p T + 685454 p^{5} T^{2} - 5876 p^{18} T^{3} + p^{34} T^{4}
7D4D_{4} 127684196T+93506538073374pT227684196p17T3+p34T4 1 - 27684196 T + 93506538073374 p T^{2} - 27684196 p^{17} T^{3} + p^{34} T^{4}
11D4D_{4} 1533184p2T3200366345168954p2T2533184p19T3+p34T4 1 - 533184 p^{2} T - 3200366345168954 p^{2} T^{2} - 533184 p^{19} T^{3} + p^{34} T^{4}
13D4D_{4} 12895838468T+1418824834264307094pT22895838468p17T3+p34T4 1 - 2895838468 T + 1418824834264307094 p T^{2} - 2895838468 p^{17} T^{3} + p^{34} T^{4}
17D4D_{4} 11580212596T+ 1 - 1580212596 T + 13 ⁣ ⁣5813\!\cdots\!58T21580212596p17T3+p34T4 T^{2} - 1580212596 p^{17} T^{3} + p^{34} T^{4}
19D4D_{4} 144213712760T+ 1 - 44213712760 T + 27 ⁣ ⁣7827\!\cdots\!78T244213712760p17T3+p34T4 T^{2} - 44213712760 p^{17} T^{3} + p^{34} T^{4}
23D4D_{4} 1487549782828T+ 1 - 487549782828 T + 24 ⁣ ⁣0224\!\cdots\!02T2487549782828p17T3+p34T4 T^{2} - 487549782828 p^{17} T^{3} + p^{34} T^{4}
29D4D_{4} 13987314863500T+ 1 - 3987314863500 T + 18 ⁣ ⁣1818\!\cdots\!18T23987314863500p17T3+p34T4 T^{2} - 3987314863500 p^{17} T^{3} + p^{34} T^{4}
31D4D_{4} 1+5492261339336T+ 1 + 5492261339336 T + 52 ⁣ ⁣4652\!\cdots\!46T2+5492261339336p17T3+p34T4 T^{2} + 5492261339336 p^{17} T^{3} + p^{34} T^{4}
37D4D_{4} 1+62715287637884T+ 1 + 62715287637884 T + 18 ⁣ ⁣9818\!\cdots\!98T2+62715287637884p17T3+p34T4 T^{2} + 62715287637884 p^{17} T^{3} + p^{34} T^{4}
41D4D_{4} 123411477277324T+ 1 - 23411477277324 T + 10 ⁣ ⁣0610\!\cdots\!06T223411477277324p17T3+p34T4 T^{2} - 23411477277324 p^{17} T^{3} + p^{34} T^{4}
43D4D_{4} 1+124856923191092T+ 1 + 124856923191092 T + 14 ⁣ ⁣0214\!\cdots\!02T2+124856923191092p17T3+p34T4 T^{2} + 124856923191092 p^{17} T^{3} + p^{34} T^{4}
47D4D_{4} 1+185946612123564T+ 1 + 185946612123564 T + 57 ⁣ ⁣9857\!\cdots\!98T2+185946612123564p17T3+p34T4 T^{2} + 185946612123564 p^{17} T^{3} + p^{34} T^{4}
53D4D_{4} 1359339780647668T+ 1 - 359339780647668 T + 33 ⁣ ⁣8233\!\cdots\!82T2359339780647668p17T3+p34T4 T^{2} - 359339780647668 p^{17} T^{3} + p^{34} T^{4}
59D4D_{4} 1902179170360600T+ 1 - 902179170360600 T + 20 ⁣ ⁣3820\!\cdots\!38T2902179170360600p17T3+p34T4 T^{2} - 902179170360600 p^{17} T^{3} + p^{34} T^{4}
61D4D_{4} 1+1564422918967676T+ 1 + 1564422918967676 T + 36 ⁣ ⁣8636\!\cdots\!86T2+1564422918967676p17T3+p34T4 T^{2} + 1564422918967676 p^{17} T^{3} + p^{34} T^{4}
67D4D_{4} 1+5839738931054684T+ 1 + 5839738931054684 T + 30 ⁣ ⁣1830\!\cdots\!18T2+5839738931054684p17T3+p34T4 T^{2} + 5839738931054684 p^{17} T^{3} + p^{34} T^{4}
71D4D_{4} 1+67588560434136T+ 1 + 67588560434136 T + 50 ⁣ ⁣0650\!\cdots\!06T2+67588560434136p17T3+p34T4 T^{2} + 67588560434136 p^{17} T^{3} + p^{34} T^{4}
73D4D_{4} 13533390699585668T+ 1 - 3533390699585668 T + 90 ⁣ ⁣6290\!\cdots\!62T23533390699585668p17T3+p34T4 T^{2} - 3533390699585668 p^{17} T^{3} + p^{34} T^{4}
79D4D_{4} 119002656396552080T+ 1 - 19002656396552080 T + 43 ⁣ ⁣1843\!\cdots\!18T219002656396552080p17T3+p34T4 T^{2} - 19002656396552080 p^{17} T^{3} + p^{34} T^{4}
83D4D_{4} 1261145638254436pT+ 1 - 261145638254436 p T + 75 ⁣ ⁣8275\!\cdots\!82T2261145638254436p18T3+p34T4 T^{2} - 261145638254436 p^{18} T^{3} + p^{34} T^{4}
89D4D_{4} 1+97499522192222220T+ 1 + 97499522192222220 T + 48 ⁣ ⁣5848\!\cdots\!58T2+97499522192222220p17T3+p34T4 T^{2} + 97499522192222220 p^{17} T^{3} + p^{34} T^{4}
97D4D_{4} 199889937855386756T+ 1 - 99889937855386756 T + 14 ⁣ ⁣5814\!\cdots\!58T299889937855386756p17T3+p34T4 T^{2} - 99889937855386756 p^{17} T^{3} + p^{34} T^{4}
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   L(s)=p j=14(1αj,pps)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

−16.49452189855450298359084947718, −15.79977185819673406160331055604, −14.91810322526225841113093940138, −14.89257222253528925539165822730, −13.87900508727161083944734268257, −13.82863554692487995426831217217, −12.66806956792330355632198207896, −11.85444941196240143799885967530, −11.21093525111772113575974931888, −10.48194131361407213808657194346, −8.716239185577053457074559797551, −8.414702524300610678224362229570, −7.56453926402556833448452837594, −6.76098960845187442525068610138, −5.11636671786388053697245566920, −4.66794664716345343582458429224, −3.48709245950675540928065591627, −3.17904426674912031285594886645, −1.90031851112469128941490957302, −1.22162989460489883800798337344, 1.22162989460489883800798337344, 1.90031851112469128941490957302, 3.17904426674912031285594886645, 3.48709245950675540928065591627, 4.66794664716345343582458429224, 5.11636671786388053697245566920, 6.76098960845187442525068610138, 7.56453926402556833448452837594, 8.414702524300610678224362229570, 8.716239185577053457074559797551, 10.48194131361407213808657194346, 11.21093525111772113575974931888, 11.85444941196240143799885967530, 12.66806956792330355632198207896, 13.82863554692487995426831217217, 13.87900508727161083944734268257, 14.89257222253528925539165822730, 14.91810322526225841113093940138, 15.79977185819673406160331055604, 16.49452189855450298359084947718

Graph of the ZZ-function along the critical line