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Properties

Label 2-312-104.11-c1-0-10
Degree $2$
Conductor $312$
Sign $0.658 + 0.752i$
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  + (−0.614 − 1.27i)2-s + (−0.5 − 0.866i)3-s + (−1.24 + 1.56i)4-s + (−2.47 + 2.47i)5-s + (−0.795 + 1.16i)6-s + (3.75 − 1.00i)7-s + (2.75 + 0.622i)8-s + (−0.499 + 0.866i)9-s + (4.68 + 1.63i)10-s + (0.509 + 0.136i)11-s + (1.97 + 0.294i)12-s + (0.521 − 3.56i)13-s + (−3.59 − 4.16i)14-s + (3.38 + 0.907i)15-s + (−0.902 − 3.89i)16-s + (5.38 + 3.10i)17-s + ⋯
L(s)  = 1  + (−0.434 − 0.900i)2-s + (−0.288 − 0.499i)3-s + (−0.622 + 0.782i)4-s + (−1.10 + 1.10i)5-s + (−0.324 + 0.477i)6-s + (1.42 − 0.380i)7-s + (0.975 + 0.220i)8-s + (−0.166 + 0.288i)9-s + (1.48 + 0.516i)10-s + (0.153 + 0.0411i)11-s + (0.571 + 0.0851i)12-s + (0.144 − 0.989i)13-s + (−0.959 − 1.11i)14-s + (0.874 + 0.234i)15-s + (−0.225 − 0.974i)16-s + (1.30 + 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.658 + 0.752i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820373 - 0.372419i\)
\(L(\frac12)\) \(\approx\) \(0.820373 - 0.372419i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.614 + 1.27i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.521 + 3.56i)T \)
good5 \( 1 + (2.47 - 2.47i)T - 5iT^{2} \)
7 \( 1 + (-3.75 + 1.00i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.509 - 0.136i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-5.38 - 3.10i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.55 + 0.953i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.37 - 2.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.839 + 0.484i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.348 - 0.348i)T - 31iT^{2} \)
37 \( 1 + (1.50 - 5.61i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.71 + 6.40i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.50 - 4.33i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.54 - 1.54i)T + 47iT^{2} \)
53 \( 1 + 14.2iT - 53T^{2} \)
59 \( 1 + (-3.44 - 12.8i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.42 - 3.13i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.76 - 6.59i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.98 + 11.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (5.69 - 5.69i)T - 73iT^{2} \)
79 \( 1 - 4.82iT - 79T^{2} \)
83 \( 1 + (10.2 + 10.2i)T + 83iT^{2} \)
89 \( 1 + (7.12 + 1.90i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.76 - 1.00i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.53794022998162405994732627732, −10.78814398199813694479678742760, −10.15190807238889201843333462328, −8.442429491598855237971786690930, −7.73544534269263641246392996645, −7.26544983989747520333062075064, −5.40419434354716416583821045828, −4.01792725091834382931584788387, −2.96671121718653164128055951097, −1.19560689739065230893976224100, 1.11986232299106953102560837281, 4.10524231057441105991970099393, 4.88189420535520903566440619166, 5.61176717758548143278437749633, 7.29025950869221681255197835949, 8.045802775207579519430607005078, 8.824380649256531402357392770538, 9.561146653415791640157891461574, 10.97244997719668348449228807772, 11.71523181684330131918615796886

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