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Properties

Label 2-464-29.5-c1-0-0
Degree $2$
Conductor $464$
Sign $0.0281 - 0.999i$
Analytic cond. $3.70505$
Root an. cond. $1.92485$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.0281 - 0.999i$
Analytic conductor: \(3.70505\)
Root analytic conductor: \(1.92485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1/2),\ 0.0281 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.181114 + 0.176086i\)
\(L(\frac12)\) \(\approx\) \(0.181114 + 0.176086i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (3.58 + 4.01i)T \)
good3 \( 1 + (2.64 - 0.603i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (2.76 + 3.47i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.416 + 1.82i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (1.37 - 2.86i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-1.39 - 0.673i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 4.31iT - 17T^{2} \)
19 \( 1 + (2.42 + 0.554i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (-3.20 + 4.02i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (4.02 - 3.20i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (-4.49 - 9.32i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 - 0.634iT - 41T^{2} \)
43 \( 1 + (-6.36 - 5.07i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-0.615 + 1.27i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (-1.81 - 2.27i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + 1.72T + 59T^{2} \)
61 \( 1 + (4.97 - 1.13i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (-9.62 + 4.63i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (9.23 + 4.44i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (2.99 + 2.38i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (-6.04 - 12.5i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (2.70 - 11.8i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (1.99 - 1.59i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (4.10 + 0.937i)T + (87.3 + 42.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line