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Properties

Label 2-476-119.10-c1-0-7
Degree $2$
Conductor $476$
Sign $0.886 + 0.463i$
Analytic cond. $3.80087$
Root an. cond. $1.94958$
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  + (0.499 + 1.01i)3-s + (0.973 − 0.853i)5-s + (−0.0472 − 2.64i)7-s + (1.04 − 1.36i)9-s + (3.17 − 0.208i)11-s + (−4.65 − 4.65i)13-s + (1.35 + 0.560i)15-s + (0.564 + 4.08i)17-s + (−0.444 − 3.37i)19-s + (2.65 − 1.37i)21-s + (2.39 + 1.18i)23-s + (−0.433 + 3.29i)25-s + (5.23 + 1.04i)27-s + (−0.486 − 2.44i)29-s + (3.32 − 1.63i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.585i)3-s + (0.435 − 0.381i)5-s + (−0.0178 − 0.999i)7-s + (0.349 − 0.455i)9-s + (0.957 − 0.0627i)11-s + (−1.29 − 1.29i)13-s + (0.349 + 0.144i)15-s + (0.137 + 0.990i)17-s + (−0.101 − 0.774i)19-s + (0.580 − 0.299i)21-s + (0.499 + 0.246i)23-s + (−0.0867 + 0.658i)25-s + (1.00 + 0.200i)27-s + (−0.0902 − 0.453i)29-s + (0.596 − 0.294i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.886 + 0.463i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ 0.886 + 0.463i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64371 - 0.404014i\)
\(L(\frac12)\) \(\approx\) \(1.64371 - 0.404014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.0472 + 2.64i)T \)
17 \( 1 + (-0.564 - 4.08i)T \)
good3 \( 1 + (-0.499 - 1.01i)T + (-1.82 + 2.38i)T^{2} \)
5 \( 1 + (-0.973 + 0.853i)T + (0.652 - 4.95i)T^{2} \)
11 \( 1 + (-3.17 + 0.208i)T + (10.9 - 1.43i)T^{2} \)
13 \( 1 + (4.65 + 4.65i)T + 13iT^{2} \)
19 \( 1 + (0.444 + 3.37i)T + (-18.3 + 4.91i)T^{2} \)
23 \( 1 + (-2.39 - 1.18i)T + (14.0 + 18.2i)T^{2} \)
29 \( 1 + (0.486 + 2.44i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (-3.32 + 1.63i)T + (18.8 - 24.5i)T^{2} \)
37 \( 1 + (0.494 - 7.53i)T + (-36.6 - 4.82i)T^{2} \)
41 \( 1 + (0.178 - 0.897i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (-3.04 - 7.35i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (0.470 - 1.75i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.31 + 3.02i)T + (-13.7 + 51.1i)T^{2} \)
59 \( 1 + (-4.63 - 0.610i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (-0.604 - 1.78i)T + (-48.3 + 37.1i)T^{2} \)
67 \( 1 + (-8.80 - 5.08i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.5 + 8.40i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (-3.01 - 1.02i)T + (57.9 + 44.4i)T^{2} \)
79 \( 1 + (1.92 - 3.89i)T + (-48.0 - 62.6i)T^{2} \)
83 \( 1 + (-4.81 + 11.6i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (14.6 + 3.92i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.0 - 2.39i)T + (89.6 - 37.1i)T^{2} \)
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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

−10.69575882451951980032502844546, −9.878300023625434031109686991765, −9.452918779400980210640601709080, −8.303691139280172067817228767016, −7.28829524001404724058105279559, −6.32713810566825514801707782994, −5.01847340401787348045572108150, −4.14174708374883462844087240000, −3.06405096526151332239678607149, −1.13676128958485638601287291603, 1.85580080269412247228129466927, 2.64781482807830311900794803281, 4.35850032488019130482347923241, 5.47064854996317059901291544905, 6.71291380327939814678966669378, 7.18081698379279137440367967155, 8.435994254770249094154077663080, 9.318302250134416736327624118753, 9.970022219869435881390037599616, 11.20140238818344960576467157370

Graph of the $Z$-function along the critical line