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Properties

Label 2-304-19.12-c4-0-31
Degree $2$
Conductor $304$
Sign $0.650 + 0.759i$
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  + (12.1 − 7.04i)3-s + (0.573 + 0.994i)5-s + 40.4·7-s + (58.6 − 101. i)9-s + 186.·11-s + (−39.2 − 22.6i)13-s + (14.0 + 8.08i)15-s + (77.4 + 134. i)17-s + (−350. − 86.3i)19-s + (493. − 284. i)21-s + (−59.5 + 103. i)23-s + (311. − 540. i)25-s − 512. i·27-s + (996. + 575. i)29-s + 258. i·31-s + ⋯
L(s)  = 1  + (1.35 − 0.782i)3-s + (0.0229 + 0.0397i)5-s + 0.825·7-s + (0.724 − 1.25i)9-s + 1.54·11-s + (−0.232 − 0.134i)13-s + (0.0622 + 0.0359i)15-s + (0.268 + 0.464i)17-s + (−0.970 − 0.239i)19-s + (1.11 − 0.646i)21-s + (−0.112 + 0.195i)23-s + (0.498 − 0.864i)25-s − 0.702i·27-s + (1.18 + 0.684i)29-s + 0.269i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.650 + 0.759i$
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: \((2,\ 304,\ (\ :2),\ 0.650 + 0.759i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.841464944\)
\(L(\frac12)\) \(\approx\) \(3.841464944\)
\(L(3)\) 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 \( 1 + (350. + 86.3i)T \)
good3 \( 1 + (-12.1 + 7.04i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-0.573 - 0.994i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 40.4T + 2.40e3T^{2} \)
11 \( 1 - 186.T + 1.46e4T^{2} \)
13 \( 1 + (39.2 + 22.6i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-77.4 - 134. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (59.5 - 103. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-996. - 575. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 258. iT - 9.23e5T^{2} \)
37 \( 1 - 368. iT - 1.87e6T^{2} \)
41 \( 1 + (-1.63e3 + 942. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.03e3 + 1.79e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (470. - 815. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (3.92e3 + 2.26e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-2.00e3 + 1.16e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.97e3 + 3.42e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-1.01e3 - 584. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (7.76e3 - 4.48e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (464. + 804. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (151. - 87.6i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 3.35e3T + 4.74e7T^{2} \)
89 \( 1 + (6.30e3 + 3.63e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-3.27e3 + 1.88e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.00741207672168231481703957943, −9.786544413487703904390085382092, −8.650992386233778505046419533893, −8.351659031494920680884330601208, −7.16446276145236254780090097992, −6.36733315641937756223730712913, −4.61370863414220915498498981347, −3.45407972881846904812960736337, −2.15117294245945255450248719938, −1.19804992517326034401753705747, 1.47282362147421678268345175579, 2.78878796254965463951335734413, 4.01834780362850392894607629472, 4.71127911085156692312359905903, 6.37250301406521285208771011172, 7.67307417283024140324891508253, 8.539978551416803280135886162732, 9.229579895681718807275891520684, 10.00244160307067280889781036945, 11.13809665584641404551880978748

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