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Properties

Label 2-312-13.9-c3-0-7
Degree $2$
Conductor $312$
Sign $0.294 - 0.955i$
Analytic cond. $18.4085$
Root an. cond. $4.29052$
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  + (−1.5 + 2.59i)3-s − 3.80·5-s + (1.42 + 2.46i)7-s + (−4.5 − 7.79i)9-s + (15.6 − 27.0i)11-s + (41.2 − 22.1i)13-s + (5.71 − 9.89i)15-s + (28.8 + 49.9i)17-s + (36.2 + 62.7i)19-s − 8.55·21-s + (−104. + 181. i)23-s − 110.·25-s + 27·27-s + (66.1 − 114. i)29-s + 275.·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s − 0.340·5-s + (0.0769 + 0.133i)7-s + (−0.166 − 0.288i)9-s + (0.428 − 0.741i)11-s + (0.880 − 0.473i)13-s + (0.0983 − 0.170i)15-s + (0.411 + 0.712i)17-s + (0.437 + 0.758i)19-s − 0.0889·21-s + (−0.950 + 1.64i)23-s − 0.884·25-s + 0.192·27-s + (0.423 − 0.733i)29-s + 1.59·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.294 - 0.955i$
Analytic conductor: \(18.4085\)
Root analytic conductor: \(4.29052\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :3/2),\ 0.294 - 0.955i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.478777681\)
\(L(\frac12)\) \(\approx\) \(1.478777681\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 - 2.59i)T \)
13 \( 1 + (-41.2 + 22.1i)T \)
good5 \( 1 + 3.80T + 125T^{2} \)
7 \( 1 + (-1.42 - 2.46i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-15.6 + 27.0i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-28.8 - 49.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-36.2 - 62.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (104. - 181. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-66.1 + 114. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 275.T + 2.97e4T^{2} \)
37 \( 1 + (183. - 317. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (84.5 - 146. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-166. - 287. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 183.T + 1.03e5T^{2} \)
53 \( 1 + 502.T + 1.48e5T^{2} \)
59 \( 1 + (-443. - 767. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-452. - 784. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-59.8 + 103. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (231. + 400. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 63.0T + 3.89e5T^{2} \)
79 \( 1 - 791.T + 4.93e5T^{2} \)
83 \( 1 - 171.T + 5.71e5T^{2} \)
89 \( 1 + (-92.6 + 160. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (363. + 630. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.72464458210058026679266943506, −10.41910633596991672251072270173, −9.758685812930291766131019274612, −8.469137601699058459444877072268, −7.85954853348905109847067333738, −6.23032294315698070947823529150, −5.62958326491212658086107412984, −4.11976472133224704896762558333, −3.29114213576716561768856084640, −1.21631317146469301885114158905, 0.66514400523474104190877882490, 2.20784380253392733155663774017, 3.85866105142486613184751582188, 4.97155576361761728720109547542, 6.32547823999604739544470122393, 7.08971770083200936151217959430, 8.113091982201021798923053889441, 9.104380580788956654970956433136, 10.21851867105843997897892351466, 11.20119813786264668315135801119

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