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Properties

Label 2-756-63.25-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.236 - 0.971i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (−0.764 + 1.32i)5-s + (−1.91 − 1.82i)7-s + (0.417 + 0.723i)11-s + (1.81 + 3.13i)13-s + (−0.301 + 0.521i)17-s + (0.846 + 1.46i)19-s + (−3.07 + 5.32i)23-s + (1.33 + 2.30i)25-s + (−4.99 + 8.65i)29-s − 3.30·31-s + (3.87 − 1.15i)35-s + (4.39 + 7.61i)37-s + (−3.51 − 6.09i)41-s + (0.846 − 1.46i)43-s − 8.46·47-s + ⋯
L(s)  = 1  + (−0.341 + 0.592i)5-s + (−0.725 − 0.688i)7-s + (0.125 + 0.218i)11-s + (0.502 + 0.870i)13-s + (−0.0730 + 0.126i)17-s + (0.194 + 0.336i)19-s + (−0.640 + 1.10i)23-s + (0.266 + 0.460i)25-s + (−0.927 + 1.60i)29-s − 0.594·31-s + (0.655 − 0.194i)35-s + (0.723 + 1.25i)37-s + (−0.549 − 0.951i)41-s + (0.129 − 0.223i)43-s − 1.23·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.236 - 0.971i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.236 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.587285 + 0.747156i\)
\(L(\frac12)\) \(\approx\) \(0.587285 + 0.747156i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (1.91 + 1.82i)T \)
good5 \( 1 + (0.764 - 1.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.417 - 0.723i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.81 - 3.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.301 - 0.521i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.846 - 1.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.07 - 5.32i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.99 - 8.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.30T + 31T^{2} \)
37 \( 1 + (-4.39 - 7.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.51 + 6.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.846 + 1.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.46T + 47T^{2} \)
53 \( 1 + (-3.99 + 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.130T + 59T^{2} \)
61 \( 1 + 4.76T + 61T^{2} \)
67 \( 1 + 2.24T + 67T^{2} \)
71 \( 1 - 9.39T + 71T^{2} \)
73 \( 1 + (2.25 - 3.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 + (-3.16 + 5.47i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.531 - 0.920i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.76 - 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.64501576344192723785164160938, −9.730986822687914805108822749864, −9.044983247220546366775170421258, −7.84087037904356593625789073115, −7.03766434543934896237341906969, −6.44843000108343415805963716180, −5.23241893182239538415662610610, −3.86036621168414099715835322268, −3.35485232113495621005798670329, −1.64214656146193573543574343418, 0.48555223655673893355612814431, 2.38542794262998263286487023901, 3.53991855369551025046908855450, 4.61791861164501097790330611484, 5.76398882279091003402622885222, 6.37565175699949634666016246896, 7.69338190111604016849666325284, 8.422361014990070773790278777455, 9.203991227962370981920175968832, 9.982718982321622428738875218584

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