Properties

Label 2-312-13.3-c1-0-5
Degree $2$
Conductor $312$
Sign $0.859 + 0.511i$
Analytic cond. $2.49133$
Root an. cond. $1.57839$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + 3·5-s + (−0.499 + 0.866i)9-s + (3.5 − 0.866i)13-s + (−1.5 − 2.59i)15-s + (0.5 − 0.866i)17-s + (−2 − 3.46i)23-s + 4·25-s + 0.999·27-s + (−1.5 − 2.59i)29-s + 8·31-s + (2.5 + 4.33i)37-s + (−2.5 − 2.59i)39-s + (−1.5 − 2.59i)41-s + (−2 + 3.46i)43-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 1.34·5-s + (−0.166 + 0.288i)9-s + (0.970 − 0.240i)13-s + (−0.387 − 0.670i)15-s + (0.121 − 0.210i)17-s + (−0.417 − 0.722i)23-s + 0.800·25-s + 0.192·27-s + (−0.278 − 0.482i)29-s + 1.43·31-s + (0.410 + 0.711i)37-s + (−0.400 − 0.416i)39-s + (−0.234 − 0.405i)41-s + (−0.304 + 0.528i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43427 - 0.394187i\)
\(L(\frac12)\) \(\approx\) \(1.43427 - 0.394187i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 13T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 12.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4 - 6.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.62292162436593495312194158447, −10.58832133759301379743022411382, −9.837914609998657231857675340173, −8.794550399354803440088808874578, −7.76901976195737233647079640152, −6.34538186879401067698037396990, −5.99007749996929647816003772971, −4.66776475550602276807200700380, −2.82895586817474621720524245291, −1.43499133906165139030918547822, 1.72159990829401135929287508890, 3.38751374332310410175321206840, 4.82361483187115110582129798381, 5.88394477089614038455528081751, 6.52662889645244401370877893506, 8.100508021663300914241474255790, 9.214684869125839839883750657889, 9.861212634020860723693032357808, 10.71888246329299168417625582925, 11.59068624652272287960495516095

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