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Properties

Label 2-312-104.77-c3-0-0
Degree $2$
Conductor $312$
Sign $-0.251 + 0.967i$
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  + (2.09 + 1.90i)2-s + 3i·3-s + (0.748 + 7.96i)4-s − 5.98·5-s + (−5.71 + 6.27i)6-s − 13.5i·7-s + (−13.5 + 18.0i)8-s − 9·9-s + (−12.5 − 11.3i)10-s − 38.8·11-s + (−23.8 + 2.24i)12-s + (−17.8 − 43.3i)13-s + (25.8 − 28.3i)14-s − 17.9i·15-s + (−62.8 + 11.9i)16-s + 10.7·17-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)2-s + 0.577i·3-s + (0.0935 + 0.995i)4-s − 0.535·5-s + (−0.388 + 0.426i)6-s − 0.732i·7-s + (−0.601 + 0.799i)8-s − 0.333·9-s + (−0.395 − 0.360i)10-s − 1.06·11-s + (−0.574 + 0.0540i)12-s + (−0.380 − 0.924i)13-s + (0.493 − 0.541i)14-s − 0.308i·15-s + (−0.982 + 0.186i)16-s + 0.153·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.251 + 0.967i$
Analytic conductor: \(18.4085\)
Root analytic conductor: \(4.29052\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :3/2),\ -0.251 + 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1244480782\)
\(L(\frac12)\) \(\approx\) \(0.1244480782\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.09 - 1.90i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (17.8 + 43.3i)T \)
good5 \( 1 + 5.98T + 125T^{2} \)
7 \( 1 + 13.5iT - 343T^{2} \)
11 \( 1 + 38.8T + 1.33e3T^{2} \)
17 \( 1 - 10.7T + 4.91e3T^{2} \)
19 \( 1 + 50.0T + 6.85e3T^{2} \)
23 \( 1 - 58.0T + 1.21e4T^{2} \)
29 \( 1 - 140. iT - 2.43e4T^{2} \)
31 \( 1 + 49.2iT - 2.97e4T^{2} \)
37 \( 1 + 109.T + 5.06e4T^{2} \)
41 \( 1 + 200. iT - 6.89e4T^{2} \)
43 \( 1 - 53.5iT - 7.95e4T^{2} \)
47 \( 1 + 95.3iT - 1.03e5T^{2} \)
53 \( 1 - 385. iT - 1.48e5T^{2} \)
59 \( 1 + 305.T + 2.05e5T^{2} \)
61 \( 1 - 164. iT - 2.26e5T^{2} \)
67 \( 1 + 962.T + 3.00e5T^{2} \)
71 \( 1 - 195. iT - 3.57e5T^{2} \)
73 \( 1 - 317. iT - 3.89e5T^{2} \)
79 \( 1 + 221.T + 4.93e5T^{2} \)
83 \( 1 + 445.T + 5.71e5T^{2} \)
89 \( 1 + 523. iT - 7.04e5T^{2} \)
97 \( 1 + 147. iT - 9.12e5T^{2} \)
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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

−12.07424253020452006498550220892, −10.92035887074862115058672029990, −10.27175664663246221832503859701, −8.827784161142077746932253293380, −7.83882254241122793346742494359, −7.21737724579657769246013019468, −5.78550910645470768211530004656, −4.88100781687235314172451408662, −3.88721531781095972585008954269, −2.83022885859425106617189839972, 0.03278526123440371894208590734, 1.90473103754356121176315009087, 2.92661086509348204200438140914, 4.35741194231641598165308782925, 5.43129390608148744842988302667, 6.45378466642846349809162095996, 7.59827788905514688186107254589, 8.736020075349112429786661706788, 9.811027140031894757204261253104, 10.90682519560621064540081877379

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