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Properties

Label 2-756-63.41-c3-0-9
Degree $2$
Conductor $756$
Sign $0.306 - 0.951i$
Analytic cond. $44.6054$
Root an. cond. $6.67873$
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  + (3.53 − 6.12i)5-s + (7.59 + 16.8i)7-s + (−7.40 + 4.27i)11-s + (45.3 + 26.1i)13-s + 38.9·17-s + 66.4i·19-s + (−173. − 100. i)23-s + (37.5 + 64.9i)25-s + (−52.9 + 30.5i)29-s + (−116. − 67.2i)31-s + (130. + 13.2i)35-s + 298.·37-s + (−221. + 383. i)41-s + (26.1 + 45.2i)43-s + (137. + 238. i)47-s + ⋯
L(s)  = 1  + (0.316 − 0.547i)5-s + (0.410 + 0.912i)7-s + (−0.202 + 0.117i)11-s + (0.967 + 0.558i)13-s + 0.555·17-s + 0.801i·19-s + (−1.57 − 0.910i)23-s + (0.300 + 0.519i)25-s + (−0.339 + 0.195i)29-s + (−0.674 − 0.389i)31-s + (0.629 + 0.0637i)35-s + 1.32·37-s + (−0.842 + 1.45i)41-s + (0.0926 + 0.160i)43-s + (0.427 + 0.740i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.306 - 0.951i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ 0.306 - 0.951i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.008962269\)
\(L(\frac12)\) \(\approx\) \(2.008962269\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-7.59 - 16.8i)T \)
good5 \( 1 + (-3.53 + 6.12i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (7.40 - 4.27i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-45.3 - 26.1i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 38.9T + 4.91e3T^{2} \)
19 \( 1 - 66.4iT - 6.85e3T^{2} \)
23 \( 1 + (173. + 100. i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (52.9 - 30.5i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (116. + 67.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 298.T + 5.06e4T^{2} \)
41 \( 1 + (221. - 383. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-26.1 - 45.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-137. - 238. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 136. iT - 1.48e5T^{2} \)
59 \( 1 + (-191. + 331. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (261. - 151. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-318. + 552. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 228. iT - 3.57e5T^{2} \)
73 \( 1 - 1.24e3iT - 3.89e5T^{2} \)
79 \( 1 + (100. + 174. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-323. - 560. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 826.T + 7.04e5T^{2} \)
97 \( 1 + (-17.0 + 9.85i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−9.961430918147853187604584707899, −9.256846340497821438120657463176, −8.389939527902547710590663326223, −7.81652722764925499554545136122, −6.34106942191923336767223621760, −5.74899328608926916759047260561, −4.77750124442991315908370495984, −3.69368244029595890865545031794, −2.26660061026440214779095441300, −1.28949379542213630938800524576, 0.57196474657375375679843338184, 1.90518880992954542451257250831, 3.27213557768260677196037092588, 4.13513689704589732191524407844, 5.40770383228414206405218669865, 6.22596631030306395709366331669, 7.26992825177128639743582147503, 7.928694241173413260382733389970, 8.898623693696083105149136424877, 10.03696653611020017360602748127

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