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Properties

Label 2-304-19.7-c3-0-11
Degree $2$
Conductor $304$
Sign $0.411 - 0.911i$
Analytic cond. $17.9365$
Root an. cond. $4.23516$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.881 + 1.52i)3-s + (10.3 + 17.9i)5-s − 8.76·7-s + (11.9 − 20.6i)9-s + 62.1·11-s + (32.2 − 55.8i)13-s + (−18.2 + 31.6i)15-s + (23.2 + 40.1i)17-s + (−17.3 + 80.9i)19-s + (−7.72 − 13.3i)21-s + (−18.6 + 32.2i)23-s + (−151. + 263. i)25-s + 89.7·27-s + (33.2 − 57.5i)29-s − 112.·31-s + ⋯
L(s)  = 1  + (0.169 + 0.293i)3-s + (0.926 + 1.60i)5-s − 0.473·7-s + (0.442 − 0.766i)9-s + 1.70·11-s + (0.688 − 1.19i)13-s + (−0.314 + 0.544i)15-s + (0.331 + 0.573i)17-s + (−0.209 + 0.977i)19-s + (−0.0802 − 0.138i)21-s + (−0.169 + 0.292i)23-s + (−1.21 + 2.10i)25-s + 0.639·27-s + (0.212 − 0.368i)29-s − 0.651·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(17.9365\)
Root analytic conductor: \(4.23516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :3/2),\ 0.411 - 0.911i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.517431787\)
\(L(\frac12)\) \(\approx\) \(2.517431787\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.3 - 80.9i)T \)
good3 \( 1 + (-0.881 - 1.52i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-10.3 - 17.9i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + 8.76T + 343T^{2} \)
11 \( 1 - 62.1T + 1.33e3T^{2} \)
13 \( 1 + (-32.2 + 55.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-23.2 - 40.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (18.6 - 32.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-33.2 + 57.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 112.T + 2.97e4T^{2} \)
37 \( 1 + 189.T + 5.06e4T^{2} \)
41 \( 1 + (-120. - 207. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-84.1 - 145. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-93.9 + 162. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-56.5 + 97.9i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-92.9 - 161. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (154. - 267. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-19.7 + 34.1i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (175. + 303. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-4.80 - 8.31i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (588. + 1.01e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 257.T + 5.71e5T^{2} \)
89 \( 1 + (-66.8 + 115. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (598. + 1.03e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.28171511819566186557110090350, −10.25634829783158185508443719070, −9.864396387591585170891230985265, −8.853351245622646083948124946025, −7.38162491758108972794911479856, −6.29083769175907439548912430494, −6.02093280573135487060972278449, −3.78589756720023524614412191945, −3.22364843856785427358423317857, −1.52205561406548328749178541437, 1.07499382615969870452995537716, 1.98616094587034710991695378790, 4.05666908103183507080386650807, 4.97954193064896548166253716803, 6.20425768533783688138603290100, 7.09117647341751178161956166949, 8.691379031288322053956515664419, 9.059684000125003294542808521271, 9.856270212130327405381025814293, 11.24598516329177225186985242842

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