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Properties

Label 2-354-177.176-c5-0-57
Degree $2$
Conductor $354$
Sign $0.954 + 0.297i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
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  + 4·2-s + (8.41 − 13.1i)3-s + 16·4-s + 13.9i·5-s + (33.6 − 52.4i)6-s − 4.42·7-s + 64·8-s + (−101. − 220. i)9-s + 55.7i·10-s + 290.·11-s + (134. − 209. i)12-s + 1.15e3i·13-s − 17.7·14-s + (182. + 117. i)15-s + 256·16-s + 1.00e3i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.540 − 0.841i)3-s + 0.5·4-s + 0.249i·5-s + (0.381 − 0.595i)6-s − 0.0341·7-s + 0.353·8-s + (−0.416 − 0.909i)9-s + 0.176i·10-s + 0.723·11-s + (0.270 − 0.420i)12-s + 1.88i·13-s − 0.0241·14-s + (0.209 + 0.134i)15-s + 0.250·16-s + 0.842i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.954 + 0.297i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ 0.954 + 0.297i)\)

Particular Values

\(L(3)\) \(\approx\) \(4.491840748\)
\(L(\frac12)\) \(\approx\) \(4.491840748\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4T \)
3 \( 1 + (-8.41 + 13.1i)T \)
59 \( 1 + (-7.09e3 - 2.57e4i)T \)
good5 \( 1 - 13.9iT - 3.12e3T^{2} \)
7 \( 1 + 4.42T + 1.68e4T^{2} \)
11 \( 1 - 290.T + 1.61e5T^{2} \)
13 \( 1 - 1.15e3iT - 3.71e5T^{2} \)
17 \( 1 - 1.00e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.30e3T + 2.47e6T^{2} \)
23 \( 1 - 3.59e3T + 6.43e6T^{2} \)
29 \( 1 + 1.83e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.54e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.59e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.38e4iT - 1.15e8T^{2} \)
43 \( 1 - 2.53e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.10e4T + 2.29e8T^{2} \)
53 \( 1 - 2.90e4iT - 4.18e8T^{2} \)
61 \( 1 + 1.00e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.51e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.96e4iT - 1.80e9T^{2} \)
73 \( 1 - 231. iT - 2.07e9T^{2} \)
79 \( 1 + 2.08e3T + 3.07e9T^{2} \)
83 \( 1 - 1.14e5T + 3.93e9T^{2} \)
89 \( 1 + 1.10e5T + 5.58e9T^{2} \)
97 \( 1 + 3.86e4iT - 8.58e9T^{2} \)
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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

−10.94675024127946023806936384730, −9.425462799268956230619404630973, −8.787415934096197329459080334108, −7.42203323155213783890517414477, −6.79148567391760127019942913249, −5.97156271983094184575529181712, −4.43179027504766770885663316246, −3.42951747376354522722975021747, −2.21568580174753201222529298010, −1.17111358771851702550987998049, 1.01016689333271027682761636941, 2.87843673977962505732258125601, 3.41576424004712814766749394455, 4.89700986372670883742441249472, 5.31701711895056543899471844519, 6.82483441912863253965392956294, 7.921528870210325738724448942827, 8.886120772517057149478060644736, 9.820717494967396413378356179927, 10.71758085112878601803740326218

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