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Properties

Label 2-464-29.16-c1-0-11
Degree 22
Conductor 464464
Sign 0.734+0.678i0.734 + 0.678i
Analytic cond. 3.705053.70505
Root an. cond. 1.924851.92485
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.52 − 1.90i)3-s + (2.60 + 1.25i)5-s + (1.89 − 2.37i)7-s + (−0.658 − 2.88i)9-s + (−1.22 + 5.34i)11-s + (0.0239 − 0.104i)13-s + (6.34 − 3.05i)15-s + 0.816·17-s + (−1.27 − 1.59i)19-s + (−1.65 − 7.24i)21-s + (−8.25 + 3.97i)23-s + (2.07 + 2.60i)25-s + (0.0938 + 0.0451i)27-s + (−5.37 + 0.304i)29-s + (−3.10 − 1.49i)31-s + ⋯
L(s)  = 1  + (0.878 − 1.10i)3-s + (1.16 + 0.559i)5-s + (0.717 − 0.899i)7-s + (−0.219 − 0.961i)9-s + (−0.367 + 1.61i)11-s + (0.00663 − 0.0290i)13-s + (1.63 − 0.789i)15-s + 0.197·17-s + (−0.292 − 0.366i)19-s + (−0.360 − 1.58i)21-s + (−1.72 + 0.829i)23-s + (0.414 + 0.520i)25-s + (0.0180 + 0.00869i)27-s + (−0.998 + 0.0564i)29-s + (−0.556 − 0.268i)31-s + ⋯

Functional equation

Λ(s)=(464s/2 ΓC(s) L(s)=((0.734+0.678i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(464s/2 ΓC(s+1/2) L(s)=((0.734+0.678i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 464464    =    24⋅292^{4} \cdot 29
Sign: 0.734+0.678i0.734 + 0.678i
Analytic conductor: 3.705053.70505
Root analytic conductor: 1.924851.92485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ464(161,⋅)\chi_{464} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 464, ( :1/2), 0.734+0.678i)(2,\ 464,\ (\ :1/2),\ 0.734 + 0.678i)

Particular Values

L(1)L(1) ≈\approx 2.10004−0.821254i2.10004 - 0.821254i
L(12)L(\frac12) ≈\approx 2.10004−0.821254i2.10004 - 0.821254i
L(32)L(\frac{3}{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}
ppFp(T)F_p(T)
bad2 1 1
29 1+(5.37−0.304i)T 1 + (5.37 - 0.304i)T
good3 1+(−1.52+1.90i)T+(−0.667−2.92i)T2 1 + (-1.52 + 1.90i)T + (-0.667 - 2.92i)T^{2}
5 1+(−2.60−1.25i)T+(3.11+3.90i)T2 1 + (-2.60 - 1.25i)T + (3.11 + 3.90i)T^{2}
7 1+(−1.89+2.37i)T+(−1.55−6.82i)T2 1 + (-1.89 + 2.37i)T + (-1.55 - 6.82i)T^{2}
11 1+(1.22−5.34i)T+(−9.91−4.77i)T2 1 + (1.22 - 5.34i)T + (-9.91 - 4.77i)T^{2}
13 1+(−0.0239+0.104i)T+(−11.7−5.64i)T2 1 + (-0.0239 + 0.104i)T + (-11.7 - 5.64i)T^{2}
17 1−0.816T+17T2 1 - 0.816T + 17T^{2}
19 1+(1.27+1.59i)T+(−4.22+18.5i)T2 1 + (1.27 + 1.59i)T + (-4.22 + 18.5i)T^{2}
23 1+(8.25−3.97i)T+(14.3−17.9i)T2 1 + (8.25 - 3.97i)T + (14.3 - 17.9i)T^{2}
31 1+(3.10+1.49i)T+(19.3+24.2i)T2 1 + (3.10 + 1.49i)T + (19.3 + 24.2i)T^{2}
37 1+(1.31+5.75i)T+(−33.3+16.0i)T2 1 + (1.31 + 5.75i)T + (-33.3 + 16.0i)T^{2}
41 1+2.43T+41T2 1 + 2.43T + 41T^{2}
43 1+(−4.94+2.37i)T+(26.8−33.6i)T2 1 + (-4.94 + 2.37i)T + (26.8 - 33.6i)T^{2}
47 1+(−1.31+5.76i)T+(−42.3−20.3i)T2 1 + (-1.31 + 5.76i)T + (-42.3 - 20.3i)T^{2}
53 1+(3.55+1.71i)T+(33.0+41.4i)T2 1 + (3.55 + 1.71i)T + (33.0 + 41.4i)T^{2}
59 1+1.13T+59T2 1 + 1.13T + 59T^{2}
61 1+(−5.09+6.38i)T+(−13.5−59.4i)T2 1 + (-5.09 + 6.38i)T + (-13.5 - 59.4i)T^{2}
67 1+(−0.212−0.932i)T+(−60.3+29.0i)T2 1 + (-0.212 - 0.932i)T + (-60.3 + 29.0i)T^{2}
71 1+(−0.531+2.32i)T+(−63.9−30.8i)T2 1 + (-0.531 + 2.32i)T + (-63.9 - 30.8i)T^{2}
73 1+(8.01−3.85i)T+(45.5−57.0i)T2 1 + (8.01 - 3.85i)T + (45.5 - 57.0i)T^{2}
79 1+(0.934+4.09i)T+(−71.1+34.2i)T2 1 + (0.934 + 4.09i)T + (-71.1 + 34.2i)T^{2}
83 1+(−9.64−12.0i)T+(−18.4+80.9i)T2 1 + (-9.64 - 12.0i)T + (-18.4 + 80.9i)T^{2}
89 1+(4.99+2.40i)T+(55.4+69.5i)T2 1 + (4.99 + 2.40i)T + (55.4 + 69.5i)T^{2}
97 1+(−7.43−9.32i)T+(−21.5+94.5i)T2 1 + (-7.43 - 9.32i)T + (-21.5 + 94.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.74220476946993983476673634276, −10.01285064936278243818754352571, −9.195569337251926700182398700538, −7.81851340327382223815877566715, −7.45200403601865274386816313552, −6.58376558866066493920567852649, −5.34363250586619089385337175724, −3.92234992552661092441227100797, −2.22689421326857065778402454036, −1.79828297139175314417048987032, 1.94295983287534860523239579831, 3.09355067933264740337005005412, 4.36433969746639456174217829797, 5.51068268022931471805444974768, 6.01276299926821425384571061737, 8.089331209950532711734438225610, 8.596159847691078721660484109427, 9.271839805039208377629551010624, 10.08688065191009583796382045234, 10.88781100498986220634827033822

Graph of the ZZ-function along the critical line