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Properties

Label 2-304-19.12-c4-0-21
Degree $2$
Conductor $304$
Sign $0.629 + 0.776i$
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  + (−13.5 + 7.80i)3-s + (14.4 + 25.0i)5-s + 26.1·7-s + (81.4 − 141. i)9-s − 151.·11-s + (25.6 + 14.8i)13-s + (−391. − 226. i)15-s + (14.4 + 25.0i)17-s + (−348. − 95.5i)19-s + (−354. + 204. i)21-s + (−116. + 201. i)23-s + (−107. + 185. i)25-s + 1.27e3i·27-s + (−1.40e3 − 813. i)29-s − 382. i·31-s + ⋯
L(s)  = 1  + (−1.50 + 0.867i)3-s + (0.579 + 1.00i)5-s + 0.534·7-s + (1.00 − 1.74i)9-s − 1.25·11-s + (0.151 + 0.0877i)13-s + (−1.74 − 1.00i)15-s + (0.0500 + 0.0866i)17-s + (−0.964 − 0.264i)19-s + (−0.803 + 0.463i)21-s + (−0.220 + 0.381i)23-s + (−0.171 + 0.297i)25-s + 1.75i·27-s + (−1.67 − 0.967i)29-s − 0.397i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.629 + 0.776i$
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: \((2,\ 304,\ (\ :2),\ 0.629 + 0.776i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4621580904\)
\(L(\frac12)\) \(\approx\) \(0.4621580904\)
\(L(3)\) 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 \( 1 + (348. + 95.5i)T \)
good3 \( 1 + (13.5 - 7.80i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-14.4 - 25.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 26.1T + 2.40e3T^{2} \)
11 \( 1 + 151.T + 1.46e4T^{2} \)
13 \( 1 + (-25.6 - 14.8i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-14.4 - 25.0i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (116. - 201. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (1.40e3 + 813. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 382. iT - 9.23e5T^{2} \)
37 \( 1 - 1.72e3iT - 1.87e6T^{2} \)
41 \( 1 + (-2.53e3 + 1.46e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-264. - 458. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.89e3 + 3.28e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.50e3 + 1.44e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-384. + 222. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.70e3 + 4.68e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-2.69e3 - 1.55e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-380. + 219. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-1.62e3 - 2.81e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-2.33e3 + 1.34e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 9.93e3T + 4.74e7T^{2} \)
89 \( 1 + (3.52e3 + 2.03e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (1.39e4 - 8.03e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.96442249472961946089952531342, −10.27897655651114809404816849883, −9.535766714225015701954389923830, −7.947644729890153008813364857383, −6.70197723853016133780605024343, −5.84505348563497975084802740484, −5.08932672634785779039946299635, −3.91073071139023368800151702143, −2.24836780547307719457484808785, −0.19954585517839840141967994530, 1.03085192440441666836896863579, 2.09196762454863703705810980143, 4.56450388034431533908704837703, 5.42772700510859530468230978099, 5.97766210285698941813405266524, 7.29706624710329238588165474160, 8.138638273509927263233958226745, 9.364785788870627527662085494740, 10.77745425036218282779492319471, 11.02468295678417959524561085020

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