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Properties

Label 2-304-19.12-c4-0-15
Degree $2$
Conductor $304$
Sign $0.448 - 0.894i$
Analytic cond. $31.4244$
Root an. cond. $5.60575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.789i)3-s + (11.3 + 19.5i)5-s + 73.1·7-s + (−39.2 + 67.9i)9-s + 110.·11-s + (−23.5 − 13.5i)13-s + (30.9 + 17.8i)15-s + (156. + 271. i)17-s + (−56.4 − 356. i)19-s + (99.9 − 57.6i)21-s + (−439. + 761. i)23-s + (56.6 − 98.1i)25-s + 251. i·27-s + (−377. − 218. i)29-s − 1.47e3i·31-s + ⋯
L(s)  = 1  + (0.151 − 0.0876i)3-s + (0.452 + 0.783i)5-s + 1.49·7-s + (−0.484 + 0.839i)9-s + 0.914·11-s + (−0.139 − 0.0804i)13-s + (0.137 + 0.0793i)15-s + (0.542 + 0.939i)17-s + (−0.156 − 0.987i)19-s + (0.226 − 0.130i)21-s + (−0.830 + 1.43i)23-s + (0.0906 − 0.157i)25-s + 0.345i·27-s + (−0.449 − 0.259i)29-s − 1.53i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.894i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.448 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.448 - 0.894i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ 0.448 - 0.894i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.663089485\)
\(L(\frac12)\) \(\approx\) \(2.663089485\)
\(L(3)\) 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 \( 1 + (56.4 + 356. i)T \)
good3 \( 1 + (-1.36 + 0.789i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-11.3 - 19.5i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 73.1T + 2.40e3T^{2} \)
11 \( 1 - 110.T + 1.46e4T^{2} \)
13 \( 1 + (23.5 + 13.5i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-156. - 271. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (439. - 761. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (377. + 218. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 1.47e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.50e3iT - 1.87e6T^{2} \)
41 \( 1 + (191. - 110. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-645. - 1.11e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (870. - 1.50e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-3.59e3 - 2.07e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-2.01e3 + 1.16e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (842. - 1.45e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (405. + 234. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (1.06e3 - 615. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-2.17e3 - 3.76e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-9.60e3 + 5.54e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 2.90e3T + 4.74e7T^{2} \)
89 \( 1 + (-415. - 240. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (1.52e4 - 8.77e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.27660030810695766440335111951, −10.46633987415487269693083885722, −9.377589644017315533825444891879, −8.187213389493498139495706216999, −7.59636698394740404487425519608, −6.27799898102602530667787648680, −5.29034382346966099185796106316, −4.06383266530920922358501222255, −2.49466826648335494654751023117, −1.50127485233946584813796255725, 0.862030113954243273410823194854, 1.97274422228694857935521073957, 3.73910260334068127436423123273, 4.87890445562671283252849998860, 5.75938874800614975893708164837, 7.05084510326910114040880571503, 8.363414740665279785493691486734, 8.855138437044405272683800481128, 9.843587963381381980479696800640, 10.99683988926889242973353998162

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