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Properties

Label 2-304-19.12-c4-0-6
Degree $2$
Conductor $304$
Sign $0.944 - 0.328i$
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  + (−3.70 + 2.13i)3-s + (−17.7 − 30.7i)5-s − 57.6·7-s + (−31.3 + 54.3i)9-s − 144.·11-s + (−165. − 95.5i)13-s + (131. + 75.9i)15-s + (273. + 474. i)17-s + (208. − 294. i)19-s + (213. − 123. i)21-s + (−97.8 + 169. i)23-s + (−318. + 551. i)25-s − 614. i·27-s + (−16.6 − 9.62i)29-s − 933. i·31-s + ⋯
L(s)  = 1  + (−0.411 + 0.237i)3-s + (−0.710 − 1.23i)5-s − 1.17·7-s + (−0.387 + 0.670i)9-s − 1.19·11-s + (−0.979 − 0.565i)13-s + (0.584 + 0.337i)15-s + (0.947 + 1.64i)17-s + (0.576 − 0.817i)19-s + (0.484 − 0.279i)21-s + (−0.184 + 0.320i)23-s + (−0.509 + 0.882i)25-s − 0.843i·27-s + (−0.0198 − 0.0114i)29-s − 0.971i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.944 - 0.328i$
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: \((2,\ 304,\ (\ :2),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5588892379\)
\(L(\frac12)\) \(\approx\) \(0.5588892379\)
\(L(3)\) 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 \( 1 + (3.70 - 2.13i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (17.7 + 30.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 57.6T + 2.40e3T^{2} \)
11 \( 1 + 144.T + 1.46e4T^{2} \)
13 \( 1 + (165. + 95.5i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-273. - 474. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (97.8 - 169. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (16.6 + 9.62i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 933. iT - 9.23e5T^{2} \)
37 \( 1 + 1.97e3iT - 1.87e6T^{2} \)
41 \( 1 + (-1.40e3 + 814. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.10e3 - 1.90e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-281. + 487. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (922. + 532. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (1.31e3 - 758. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.67e3 - 4.63e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-4.36e3 - 2.52e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-1.23e3 + 711. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-2.82e3 - 4.88e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (532. - 307. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 2.74e3T + 4.74e7T^{2} \)
89 \( 1 + (-4.64e3 - 2.68e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (8.50e3 - 4.90e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.07399193756846923965984453313, −10.19828828083597863258003518298, −9.338914092275137445459553706431, −8.102982209524743749723722445533, −7.57684640648430644227935712444, −5.82366341440669905417311265469, −5.20379904432016484510160096531, −4.03399312737260105232293284279, −2.64471662110208763885974285005, −0.55527618394864401964126925362, 0.35492868940925517935414306453, 2.82398985228923126566885804610, 3.34014700293943149847105192097, 5.10131538972768278447949387796, 6.31061157357150152595349496356, 7.10223177911427309210609120509, 7.78970871322344167268143536879, 9.440278313773927341679936522058, 10.08674931623405069647925997281, 11.09971550277450402530640207084

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