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Properties

Label 2-304-19.12-c4-0-37
Degree $2$
Conductor $304$
Sign $-0.944 + 0.327i$
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  + (11.3 − 6.53i)3-s + (−19.8 − 34.3i)5-s + 76.8·7-s + (44.9 − 77.8i)9-s − 167.·11-s + (−266. − 153. i)13-s + (−449. − 259. i)15-s + (73.9 + 128. i)17-s + (208. − 294. i)19-s + (870. − 502. i)21-s + (−22.2 + 38.4i)23-s + (−474. + 821. i)25-s − 116. i·27-s + (608. + 351. i)29-s − 596. i·31-s + ⋯
L(s)  = 1  + (1.25 − 0.726i)3-s + (−0.793 − 1.37i)5-s + 1.56·7-s + (0.555 − 0.961i)9-s − 1.38·11-s + (−1.57 − 0.909i)13-s + (−1.99 − 1.15i)15-s + (0.256 + 0.443i)17-s + (0.577 − 0.816i)19-s + (1.97 − 1.13i)21-s + (−0.0419 + 0.0727i)23-s + (−0.759 + 1.31i)25-s − 0.160i·27-s + (0.723 + 0.417i)29-s − 0.620i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.944 + 0.327i$
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: \((2,\ 304,\ (\ :2),\ -0.944 + 0.327i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.175938947\)
\(L(\frac12)\) \(\approx\) \(2.175938947\)
\(L(3)\) 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 \( 1 + (-208. + 294. i)T \)
good3 \( 1 + (-11.3 + 6.53i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (19.8 + 34.3i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 76.8T + 2.40e3T^{2} \)
11 \( 1 + 167.T + 1.46e4T^{2} \)
13 \( 1 + (266. + 153. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-73.9 - 128. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (22.2 - 38.4i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-608. - 351. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 596. iT - 9.23e5T^{2} \)
37 \( 1 - 300. iT - 1.87e6T^{2} \)
41 \( 1 + (103. - 59.5i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (338. + 585. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (942. - 1.63e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (572. + 330. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (3.73e3 - 2.15e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-3.35e3 + 5.81e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (5.77e3 + 3.33e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-7.20e3 + 4.16e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (3.70e3 + 6.42e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-5.66e3 + 3.27e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.52e3T + 4.74e7T^{2} \)
89 \( 1 + (1.12e3 + 647. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (5.99e3 - 3.46e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.69870907718254004989329790370, −9.371155326463896157013607124839, −8.356285291491911878501208251803, −7.84860050753675618425989797453, −7.54387599171577971035316636645, −5.14718588187805122511020611863, −4.74074362353428190114084732206, −2.98971523207650835040203670503, −1.83262781035746871654919973328, −0.53557997244119912990726093936, 2.23979595044993549384565851073, 2.98442335449274180320285094617, 4.23049365215595421908117566390, 5.12606867290286573269138540469, 7.16657965683562976726914913684, 7.76669460371442105926654115411, 8.407196197215306785959861932413, 9.803275666455004863069505128179, 10.41001261185085358241036638079, 11.38503152259653801088154435207

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