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Properties

Label 2-304-19.12-c4-0-2
Degree $2$
Conductor $304$
Sign $-0.525 - 0.850i$
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  + (1.61 − 0.934i)3-s + (−20.3 − 35.1i)5-s + 11.8·7-s + (−38.7 + 67.1i)9-s + 91.6·11-s + (−81.2 − 46.8i)13-s + (−65.7 − 37.9i)15-s + (−205. − 356. i)17-s + (−333. − 139. i)19-s + (19.2 − 11.0i)21-s + (168. − 291. i)23-s + (−512. + 888. i)25-s + 296. i·27-s + (932. + 538. i)29-s + 1.78e3i·31-s + ⋯
L(s)  = 1  + (0.179 − 0.103i)3-s + (−0.812 − 1.40i)5-s + 0.242·7-s + (−0.478 + 0.828i)9-s + 0.757·11-s + (−0.480 − 0.277i)13-s + (−0.292 − 0.168i)15-s + (−0.711 − 1.23i)17-s + (−0.922 − 0.385i)19-s + (0.0435 − 0.0251i)21-s + (0.318 − 0.551i)23-s + (−0.820 + 1.42i)25-s + 0.406i·27-s + (1.10 + 0.639i)29-s + 1.85i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.525 - 0.850i$
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: \((2,\ 304,\ (\ :2),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1675490283\)
\(L(\frac12)\) \(\approx\) \(0.1675490283\)
\(L(3)\) 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 \( 1 + (333. + 139. i)T \)
good3 \( 1 + (-1.61 + 0.934i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (20.3 + 35.1i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 11.8T + 2.40e3T^{2} \)
11 \( 1 - 91.6T + 1.46e4T^{2} \)
13 \( 1 + (81.2 + 46.8i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (205. + 356. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-168. + 291. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-932. - 538. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.78e3iT - 9.23e5T^{2} \)
37 \( 1 - 240. iT - 1.87e6T^{2} \)
41 \( 1 + (1.87e3 - 1.08e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.14e3 - 1.97e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (818. - 1.41e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-108. - 62.8i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (3.42e3 - 1.97e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.09e3 - 1.89e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (1.58e3 + 915. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (994. - 574. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (2.47e3 + 4.28e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (103. - 60.0i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.03e4T + 4.74e7T^{2} \)
89 \( 1 + (661. + 381. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-3.75e3 + 2.16e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.55023476298260027289274937777, −10.59847357223486270937662936680, −9.115764352312289674534835473272, −8.627735764189498806834513318383, −7.77881940904150321550016315186, −6.62879260834910991183546153345, −4.81446449320483036164176742113, −4.74427104957324348925588440872, −2.92602777922697758983070092300, −1.31312816157553080086446328196, 0.05152802100325032486428545982, 2.18352769466654842355088321675, 3.51346789188946771431916834369, 4.21197265962153913889575104809, 6.15546057910658287568450613757, 6.73897713766315757069610886207, 7.87062727396382254416014043361, 8.784605919087445967444870263038, 9.931870668049583449263224442662, 10.88921459041833799663434157115

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