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Properties

Label 2-312-13.4-c1-0-4
Degree $2$
Conductor $312$
Sign $0.252 + 0.967i$
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 − 1.73i·5-s + (2.36 − 1.36i)7-s + (−0.499 − 0.866i)9-s + (−2.36 − 1.36i)11-s + (−2.59 + 2.5i)13-s + (−1.49 − 0.866i)15-s + (−0.133 − 0.232i)17-s + (4.09 − 2.36i)19-s − 2.73i·21-s + (4.09 − 7.09i)23-s + 2.00·25-s − 0.999·27-s + (−3.96 + 6.86i)29-s + 1.46i·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s − 0.774i·5-s + (0.894 − 0.516i)7-s + (−0.166 − 0.288i)9-s + (−0.713 − 0.411i)11-s + (−0.720 + 0.693i)13-s + (−0.387 − 0.223i)15-s + (−0.0324 − 0.0562i)17-s + (0.940 − 0.542i)19-s − 0.596i·21-s + (0.854 − 1.48i)23-s + 0.400·25-s − 0.192·27-s + (−0.736 + 1.27i)29-s + 0.262i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14696 - 0.885954i\)
\(L(\frac12)\) \(\approx\) \(1.14696 - 0.885954i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 2.5i)T \)
good5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 + (-2.36 + 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.36 + 1.36i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.133 + 0.232i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.09 + 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.09 + 7.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.96 - 6.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (1.33 + 0.767i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.33 - 2.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.09 - 10.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.26iT - 47T^{2} \)
53 \( 1 + 7.92T + 53T^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.13 - 5.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.56 - 4.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.90 - 1.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.19iT - 73T^{2} \)
79 \( 1 + 8.39T + 79T^{2} \)
83 \( 1 - 1.66iT - 83T^{2} \)
89 \( 1 + (-8.19 - 4.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.66 + 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.46099924505543795128246299006, −10.73265660904626181888078084806, −9.381647637432627623178919736172, −8.597216263978138746535577532523, −7.68293681740288811376459245284, −6.84720356521247915658918832811, −5.26065510594354946772223595414, −4.52207681409553391408178808332, −2.75927663984589777926092830469, −1.14072775553396600938989915699, 2.24902829932421009545389787883, 3.40098885423237615343708825974, 4.93386284376979485275501501964, 5.68838040704008281169977780597, 7.39828777306652560919909187597, 7.87062211457166473083539087130, 9.195822822265148108804798509575, 10.03395778297548306140089597367, 10.91175525524969068327761210202, 11.67186455309041888887633868341

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