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Properties

Label 2-304-19.7-c5-0-28
Degree $2$
Conductor $304$
Sign $0.553 - 0.832i$
Analytic cond. $48.7566$
Root an. cond. $6.98259$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.99 + 6.92i)3-s + (31.6 + 54.7i)5-s + 80.2·7-s + (89.5 − 155. i)9-s + 475.·11-s + (−337. + 584. i)13-s + (−252. + 437. i)15-s + (866. + 1.50e3i)17-s + (574. − 1.46e3i)19-s + (320. + 555. i)21-s + (2.42e3 − 4.19e3i)23-s + (−435. + 754. i)25-s + 3.37e3·27-s + (2.39e3 − 4.14e3i)29-s + 127.·31-s + ⋯
L(s)  = 1  + (0.256 + 0.444i)3-s + (0.565 + 0.979i)5-s + 0.619·7-s + (0.368 − 0.638i)9-s + 1.18·11-s + (−0.553 + 0.959i)13-s + (−0.289 + 0.502i)15-s + (0.727 + 1.25i)17-s + (0.365 − 0.931i)19-s + (0.158 + 0.274i)21-s + (0.955 − 1.65i)23-s + (−0.139 + 0.241i)25-s + 0.890·27-s + (0.528 − 0.915i)29-s + 0.0237·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(48.7566\)
Root analytic conductor: \(6.98259\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :5/2),\ 0.553 - 0.832i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.247461805\)
\(L(\frac12)\) \(\approx\) \(3.247461805\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-574. + 1.46e3i)T \)
good3 \( 1 + (-3.99 - 6.92i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-31.6 - 54.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 - 80.2T + 1.68e4T^{2} \)
11 \( 1 - 475.T + 1.61e5T^{2} \)
13 \( 1 + (337. - 584. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-866. - 1.50e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (-2.42e3 + 4.19e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-2.39e3 + 4.14e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 - 127.T + 2.86e7T^{2} \)
37 \( 1 + 1.39e4T + 6.93e7T^{2} \)
41 \( 1 + (-7.88e3 - 1.36e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (964. + 1.66e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (8.09e3 - 1.40e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-8.92e3 + 1.54e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (8.42e3 + 1.45e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.16e4 + 2.01e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.36e4 - 2.35e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-3.74e4 - 6.48e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (-3.49e4 - 6.04e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.14e4 + 1.97e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 5.80e4T + 3.93e9T^{2} \)
89 \( 1 + (-8.20e3 + 1.42e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (-6.57e4 - 1.13e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−10.93050977988238976582716463618, −10.02119047516016626193858751858, −9.305947095110584317093279135667, −8.327225193976851480361609240023, −6.76726159415583397564823981984, −6.49625294131101923390314826082, −4.80059861318094465233774957875, −3.81643955700153674977989579146, −2.56256614339299967012516869530, −1.23047015058913751721594322106, 1.04341915670924118842273705649, 1.70914007642731485798345192246, 3.33145619888897442337663606992, 4.98902332191193000856572337447, 5.41651743239430578135878554900, 7.09412859203820168453173660435, 7.79564866047333923913790973506, 8.884835767010764947844749468313, 9.620280602875824622981207744495, 10.67515839106875594215194411731

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