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Properties

Label 2-464-29.5-c1-0-0
Degree 22
Conductor 464464
Sign 0.0281−0.999i0.0281 - 0.999i
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  + (−2.64 + 0.603i)3-s + (−2.76 − 3.47i)5-s + (−0.416 − 1.82i)7-s + (3.92 − 1.88i)9-s + (−1.37 + 2.86i)11-s + (1.39 + 0.673i)13-s + (9.41 + 7.50i)15-s + 4.31i·17-s + (−2.42 − 0.554i)19-s + (2.20 + 4.57i)21-s + (3.20 − 4.02i)23-s + (−3.27 + 14.3i)25-s + (−2.86 + 2.28i)27-s + (−3.58 − 4.01i)29-s + (−4.02 + 3.20i)31-s + ⋯
L(s)  = 1  + (−1.52 + 0.348i)3-s + (−1.23 − 1.55i)5-s + (−0.157 − 0.690i)7-s + (1.30 − 0.629i)9-s + (−0.415 + 0.863i)11-s + (0.387 + 0.186i)13-s + (2.43 + 1.93i)15-s + 1.04i·17-s + (−0.557 − 0.127i)19-s + (0.480 + 0.998i)21-s + (0.669 − 0.839i)23-s + (−0.655 + 2.87i)25-s + (−0.551 + 0.440i)27-s + (−0.665 − 0.746i)29-s + (−0.722 + 0.575i)31-s + ⋯

Functional equation

Λ(s)=(464s/2 ΓC(s) L(s)=((0.0281−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0281 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(464s/2 ΓC(s+1/2) L(s)=((0.0281−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0281 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 464464    =    24⋅292^{4} \cdot 29
Sign: 0.0281−0.999i0.0281 - 0.999i
Analytic conductor: 3.705053.70505
Root analytic conductor: 1.924851.92485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ464(353,⋅)\chi_{464} (353, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 464, ( :1/2), 0.0281−0.999i)(2,\ 464,\ (\ :1/2),\ 0.0281 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.181114+0.176086i0.181114 + 0.176086i
L(12)L(\frac12) ≈\approx 0.181114+0.176086i0.181114 + 0.176086i
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+(3.58+4.01i)T 1 + (3.58 + 4.01i)T
good3 1+(2.64−0.603i)T+(2.70−1.30i)T2 1 + (2.64 - 0.603i)T + (2.70 - 1.30i)T^{2}
5 1+(2.76+3.47i)T+(−1.11+4.87i)T2 1 + (2.76 + 3.47i)T + (-1.11 + 4.87i)T^{2}
7 1+(0.416+1.82i)T+(−6.30+3.03i)T2 1 + (0.416 + 1.82i)T + (-6.30 + 3.03i)T^{2}
11 1+(1.37−2.86i)T+(−6.85−8.60i)T2 1 + (1.37 - 2.86i)T + (-6.85 - 8.60i)T^{2}
13 1+(−1.39−0.673i)T+(8.10+10.1i)T2 1 + (-1.39 - 0.673i)T + (8.10 + 10.1i)T^{2}
17 1−4.31iT−17T2 1 - 4.31iT - 17T^{2}
19 1+(2.42+0.554i)T+(17.1+8.24i)T2 1 + (2.42 + 0.554i)T + (17.1 + 8.24i)T^{2}
23 1+(−3.20+4.02i)T+(−5.11−22.4i)T2 1 + (-3.20 + 4.02i)T + (-5.11 - 22.4i)T^{2}
31 1+(4.02−3.20i)T+(6.89−30.2i)T2 1 + (4.02 - 3.20i)T + (6.89 - 30.2i)T^{2}
37 1+(−4.49−9.32i)T+(−23.0+28.9i)T2 1 + (-4.49 - 9.32i)T + (-23.0 + 28.9i)T^{2}
41 1−0.634iT−41T2 1 - 0.634iT - 41T^{2}
43 1+(−6.36−5.07i)T+(9.56+41.9i)T2 1 + (-6.36 - 5.07i)T + (9.56 + 41.9i)T^{2}
47 1+(−0.615+1.27i)T+(−29.3−36.7i)T2 1 + (-0.615 + 1.27i)T + (-29.3 - 36.7i)T^{2}
53 1+(−1.81−2.27i)T+(−11.7+51.6i)T2 1 + (-1.81 - 2.27i)T + (-11.7 + 51.6i)T^{2}
59 1+1.72T+59T2 1 + 1.72T + 59T^{2}
61 1+(4.97−1.13i)T+(54.9−26.4i)T2 1 + (4.97 - 1.13i)T + (54.9 - 26.4i)T^{2}
67 1+(−9.62+4.63i)T+(41.7−52.3i)T2 1 + (-9.62 + 4.63i)T + (41.7 - 52.3i)T^{2}
71 1+(9.23+4.44i)T+(44.2+55.5i)T2 1 + (9.23 + 4.44i)T + (44.2 + 55.5i)T^{2}
73 1+(2.99+2.38i)T+(16.2+71.1i)T2 1 + (2.99 + 2.38i)T + (16.2 + 71.1i)T^{2}
79 1+(−6.04−12.5i)T+(−49.2+61.7i)T2 1 + (-6.04 - 12.5i)T + (-49.2 + 61.7i)T^{2}
83 1+(2.70−11.8i)T+(−74.7−36.0i)T2 1 + (2.70 - 11.8i)T + (-74.7 - 36.0i)T^{2}
89 1+(1.99−1.59i)T+(19.8−86.7i)T2 1 + (1.99 - 1.59i)T + (19.8 - 86.7i)T^{2}
97 1+(4.10+0.937i)T+(87.3+42.0i)T2 1 + (4.10 + 0.937i)T + (87.3 + 42.0i)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

−11.22030888283354360026279028352, −10.67329231662761304314224696524, −9.599898785873754061375354815290, −8.505936185410698297435576586006, −7.60545314868082586091742748954, −6.53051853684108800981000836175, −5.35817130536667835211416316463, −4.49500205117024537531609442254, −4.03609565320221315102311228551, −1.10622025948825749656510278929, 0.22971700805317279590957850962, 2.74442721685078068706152809363, 3.89757720792399578515126337653, 5.44057002329328501105085731341, 6.10515292076300843200780259168, 7.11521009979473403258563180801, 7.64337705640900057984286988316, 9.026365578145629254109520087179, 10.52188279715541001184358254705, 11.05356378744306404137890423417

Graph of the ZZ-function along the critical line