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Properties

Label 4-14e4-1.1-c9e2-0-1
Degree 44
Conductor 3841638416
Sign 11
Analytic cond. 10190.310190.3
Root an. cond. 10.047210.0472
Motivic weight 99
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  − 228·3-s + 666·5-s + 1.96e4·9-s + 3.04e4·11-s − 6.46e4·13-s − 1.51e5·15-s − 5.90e5·17-s − 3.46e4·19-s − 1.04e6·23-s + 1.95e6·25-s − 1.61e6·27-s + 8.81e6·29-s + 7.40e6·31-s − 6.93e6·33-s − 1.02e7·37-s + 1.47e7·39-s + 3.67e7·41-s − 5.04e5·43-s + 1.31e7·45-s + 4.95e7·47-s + 1.34e8·51-s + 6.63e7·53-s + 2.02e7·55-s + 7.90e6·57-s + 6.15e7·59-s − 3.56e7·61-s − 4.30e7·65-s + ⋯
L(s)  = 1  − 1.62·3-s + 0.476·5-s + 9-s + 0.626·11-s − 0.628·13-s − 0.774·15-s − 1.71·17-s − 0.0610·19-s − 0.781·23-s + 25-s − 0.583·27-s + 2.31·29-s + 1.43·31-s − 1.01·33-s − 0.897·37-s + 1.02·39-s + 2.02·41-s − 0.0225·43-s + 0.476·45-s + 1.48·47-s + 2.78·51-s + 1.15·53-s + 0.298·55-s + 0.0992·57-s + 0.661·59-s − 0.329·61-s − 0.299·65-s + ⋯

Functional equation

Λ(s)=(38416s/2 ΓC(s)2 L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(38416s/2 ΓC(s+9/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3841638416    =    24⋅742^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 10190.310190.3
Root analytic conductor: 10.047210.0472
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 38416, ( :9/2,9/2), 1)(4,\ 38416,\ (\ :9/2, 9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 1.9756544001.975654400
L(12)L(\frac12) ≈\approx 1.9756544001.975654400
L(112)L(\frac{11}{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)
bad2 1 1
7 1 1
good3C22C_2^2 1+76pT+3589p2T2+76p10T3+p18T4 1 + 76 p T + 3589 p^{2} T^{2} + 76 p^{10} T^{3} + p^{18} T^{4}
5C22C_2^2 1−666T−1509569T2−666p9T3+p18T4 1 - 666 T - 1509569 T^{2} - 666 p^{9} T^{3} + p^{18} T^{4}
11C22C_2^2 1−30420T−1432571291T2−30420p9T3+p18T4 1 - 30420 T - 1432571291 T^{2} - 30420 p^{9} T^{3} + p^{18} T^{4}
13C2C_2 (1+32338T+p9T2)2 ( 1 + 32338 T + p^{9} T^{2} )^{2}
17C22C_2^2 1+590994T+230686031539T2+590994p9T3+p18T4 1 + 590994 T + 230686031539 T^{2} + 590994 p^{9} T^{3} + p^{18} T^{4}
19C22C_2^2 1+34676T−321485272803T2+34676p9T3+p18T4 1 + 34676 T - 321485272803 T^{2} + 34676 p^{9} T^{3} + p^{18} T^{4}
23C22C_2^2 1+1048536T−701724918167T2+1048536p9T3+p18T4 1 + 1048536 T - 701724918167 T^{2} + 1048536 p^{9} T^{3} + p^{18} T^{4}
29C2C_2 (1−4409406T+p9T2)2 ( 1 - 4409406 T + p^{9} T^{2} )^{2}
31C22C_2^2 1−7401184T+28337902441185T2−7401184p9T3+p18T4 1 - 7401184 T + 28337902441185 T^{2} - 7401184 p^{9} T^{3} + p^{18} T^{4}
37C22C_2^2 1+10234502T−25216708607073T2+10234502p9T3+p18T4 1 + 10234502 T - 25216708607073 T^{2} + 10234502 p^{9} T^{3} + p^{18} T^{4}
41C2C_2 (1−18352746T+p9T2)2 ( 1 - 18352746 T + p^{9} T^{2} )^{2}
43C2C_2 (1+252340T+p9T2)2 ( 1 + 252340 T + p^{9} T^{2} )^{2}
47C22C_2^2 1−49517136T+1332816284539729T2−49517136p9T3+p18T4 1 - 49517136 T + 1332816284539729 T^{2} - 49517136 p^{9} T^{3} + p^{18} T^{4}
53C22C_2^2 1−66396906T+1108785534570703T2−66396906p9T3+p18T4 1 - 66396906 T + 1108785534570703 T^{2} - 66396906 p^{9} T^{3} + p^{18} T^{4}
59C22C_2^2 1−61523748T−4877824250687435T2−61523748p9T3+p18T4 1 - 61523748 T - 4877824250687435 T^{2} - 61523748 p^{9} T^{3} + p^{18} T^{4}
61C22C_2^2 1+35638622T−10424034714775257T2+35638622p9T3+p18T4 1 + 35638622 T - 10424034714775257 T^{2} + 35638622 p^{9} T^{3} + p^{18} T^{4}
67C22C_2^2 1+181742372T+5823755383891437T2+181742372p9T3+p18T4 1 + 181742372 T + 5823755383891437 T^{2} + 181742372 p^{9} T^{3} + p^{18} T^{4}
71C2C_2 (1−90904968T+p9T2)2 ( 1 - 90904968 T + p^{9} T^{2} )^{2}
73C22C_2^2 1−262978678T+10286198374359771T2−262978678p9T3+p18T4 1 - 262978678 T + 10286198374359771 T^{2} - 262978678 p^{9} T^{3} + p^{18} T^{4}
79C22C_2^2 1−116502832T−106278686118598095T2−116502832p9T3+p18T4 1 - 116502832 T - 106278686118598095 T^{2} - 116502832 p^{9} T^{3} + p^{18} T^{4}
83C2C_2 (1+9563724T+p9T2)2 ( 1 + 9563724 T + p^{9} T^{2} )^{2}
89C22C_2^2 1+611826714T+23975524256552587T2+611826714p9T3+p18T4 1 + 611826714 T + 23975524256552587 T^{2} + 611826714 p^{9} T^{3} + p^{18} T^{4}
97C2C_2 (1+259312798T+p9T2)2 ( 1 + 259312798 T + p^{9} T^{2} )^{2}
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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

−10.92885158683998848690320456234, −10.75335532318475153533060713557, −10.32555116267252232845906795820, −9.642128856924937169747340041343, −9.286018628773007293070383461082, −8.530415950199590885272733694587, −8.263985450927726004990980026538, −7.24746930420000248706995066983, −6.77554037916555869637461693035, −6.49531502461069077966208174344, −5.89160763118829976715389922641, −5.56365806197580868428675299401, −4.65298807266619584120924875624, −4.58635887526713798532705816180, −3.87888115194624231126762499464, −2.65945268162743583615632631211, −2.44863574370038809959705020693, −1.51794658397927554468257978903, −0.64927693097544580724205138325, −0.56630292061632817830660247961, 0.56630292061632817830660247961, 0.64927693097544580724205138325, 1.51794658397927554468257978903, 2.44863574370038809959705020693, 2.65945268162743583615632631211, 3.87888115194624231126762499464, 4.58635887526713798532705816180, 4.65298807266619584120924875624, 5.56365806197580868428675299401, 5.89160763118829976715389922641, 6.49531502461069077966208174344, 6.77554037916555869637461693035, 7.24746930420000248706995066983, 8.263985450927726004990980026538, 8.530415950199590885272733694587, 9.286018628773007293070383461082, 9.642128856924937169747340041343, 10.32555116267252232845906795820, 10.75335532318475153533060713557, 10.92885158683998848690320456234

Graph of the ZZ-function along the critical line