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Properties

Label 2-304-19.12-c4-0-13
Degree $2$
Conductor $304$
Sign $0.824 + 0.565i$
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  + (−5.09 + 2.94i)3-s + (0.570 + 0.988i)5-s − 80.7·7-s + (−23.1 + 40.1i)9-s − 9.86·11-s + (−9.59 − 5.54i)13-s + (−5.82 − 3.36i)15-s + (52.8 + 91.5i)17-s + (−360. + 7.47i)19-s + (411. − 237. i)21-s + (−40.9 + 70.9i)23-s + (311. − 540. i)25-s − 749. i·27-s + (753. + 435. i)29-s + 643. i·31-s + ⋯
L(s)  = 1  + (−0.566 + 0.327i)3-s + (0.0228 + 0.0395i)5-s − 1.64·7-s + (−0.285 + 0.495i)9-s − 0.0814·11-s + (−0.0567 − 0.0327i)13-s + (−0.0258 − 0.0149i)15-s + (0.182 + 0.316i)17-s + (−0.999 + 0.0207i)19-s + (0.933 − 0.538i)21-s + (−0.0774 + 0.134i)23-s + (0.498 − 0.864i)25-s − 1.02i·27-s + (0.896 + 0.517i)29-s + 0.669i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.824 + 0.565i$
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: \((2,\ 304,\ (\ :2),\ 0.824 + 0.565i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6994327954\)
\(L(\frac12)\) \(\approx\) \(0.6994327954\)
\(L(3)\) 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 \( 1 + (360. - 7.47i)T \)
good3 \( 1 + (5.09 - 2.94i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-0.570 - 0.988i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 80.7T + 2.40e3T^{2} \)
11 \( 1 + 9.86T + 1.46e4T^{2} \)
13 \( 1 + (9.59 + 5.54i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-52.8 - 91.5i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (40.9 - 70.9i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-753. - 435. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 643. iT - 9.23e5T^{2} \)
37 \( 1 + 1.26e3iT - 1.87e6T^{2} \)
41 \( 1 + (317. - 183. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (744. + 1.29e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (614. - 1.06e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.52e3 - 1.45e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-2.50e3 + 1.44e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-996. + 1.72e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.07e3 - 1.77e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-7.03e3 + 4.06e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (3.13e3 + 5.42e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-9.96e3 + 5.75e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 4.64e3T + 4.74e7T^{2} \)
89 \( 1 + (5.01e3 + 2.89e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-3.79e3 + 2.19e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.65852756915612346308886728999, −10.32816644194421923232915784673, −9.227264498420574097146417793035, −8.212247337053420664562868524074, −6.79061532594356027458477446669, −6.12424418811787458934414787689, −4.99940725679707608526448476581, −3.70105839059623972240768213732, −2.47950070672528364917732028055, −0.34977640919656947897477565362, 0.73936575487318825358492968055, 2.70177786522626701872479976420, 3.81596900772606622188475104324, 5.38291593855000913830429102044, 6.44200157730843639525952424545, 6.86216765666105055075343587734, 8.385420587622150970715390431544, 9.433668150713650956239698081143, 10.13905838693385031073460150520, 11.28622597795747458921829225124

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