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Properties

Label 2-304-19.7-c5-0-45
Degree $2$
Conductor $304$
Sign $-0.682 + 0.731i$
Analytic cond. $48.7566$
Root an. cond. $6.98259$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.30 + 5.72i)3-s + (12.6 + 21.8i)5-s + 40.7·7-s + (99.6 − 172. i)9-s − 324.·11-s + (−48.8 + 84.6i)13-s + (−83.3 + 144. i)15-s + (−1.05e3 − 1.83e3i)17-s + (−806. + 1.35e3i)19-s + (134. + 232. i)21-s + (−1.50e3 + 2.60e3i)23-s + (1.24e3 − 2.15e3i)25-s + 2.92e3·27-s + (−527. + 914. i)29-s − 7.60e3·31-s + ⋯
L(s)  = 1  + (0.211 + 0.367i)3-s + (0.225 + 0.390i)5-s + 0.314·7-s + (0.410 − 0.710i)9-s − 0.809·11-s + (−0.0801 + 0.138i)13-s + (−0.0955 + 0.165i)15-s + (−0.887 − 1.53i)17-s + (−0.512 + 0.858i)19-s + (0.0665 + 0.115i)21-s + (−0.593 + 1.02i)23-s + (0.398 − 0.689i)25-s + 0.771·27-s + (−0.116 + 0.201i)29-s − 1.42·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.731i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.682 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.682 + 0.731i$
Analytic conductor: \(48.7566\)
Root analytic conductor: \(6.98259\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :5/2),\ -0.682 + 0.731i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5459910534\)
\(L(\frac12)\) \(\approx\) \(0.5459910534\)
\(L(\frac{7}{2})\) 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 \( 1 + (806. - 1.35e3i)T \)
good3 \( 1 + (-3.30 - 5.72i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-12.6 - 21.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 - 40.7T + 1.68e4T^{2} \)
11 \( 1 + 324.T + 1.61e5T^{2} \)
13 \( 1 + (48.8 - 84.6i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (1.05e3 + 1.83e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (1.50e3 - 2.60e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (527. - 914. i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + 7.60e3T + 2.86e7T^{2} \)
37 \( 1 + 2.05e3T + 6.93e7T^{2} \)
41 \( 1 + (-230. - 399. i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-188. - 326. i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-8.35e3 + 1.44e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-5.23e3 + 9.06e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (2.37e4 + 4.11e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.37e4 - 2.37e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.72e4 - 2.98e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (1.87e3 + 3.25e3i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (-6.14e3 - 1.06e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.67e4 + 4.62e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.31e3T + 3.93e9T^{2} \)
89 \( 1 + (-5.17e4 + 8.96e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (5.29e4 + 9.17e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−10.43790445842628978337996812257, −9.647083691877555458890722064163, −8.777074562658561731815976152465, −7.58107125671497592038813242341, −6.69093183525171171948391439336, −5.45925839062771723477391165566, −4.34137369372075044436773359066, −3.15731780410240033048917161364, −1.89986052989185577436389145030, −0.12914676380206150994374138560, 1.56842241896837183568690314595, 2.49729266772358759987882702129, 4.22849443468866907331006503240, 5.16399172813402949256055720550, 6.36853782785318513886931237162, 7.52630359526646026603258994123, 8.322779403182795683058712841172, 9.168804916320467813267687056031, 10.61615276810527350796463491388, 10.86345833014588451928621495022

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