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Properties

Label 2-756-63.41-c3-0-19
Degree $2$
Conductor $756$
Sign $0.346 + 0.937i$
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  + (2.20 − 3.82i)5-s + (13.5 − 12.6i)7-s + (59.1 − 34.1i)11-s + (29.9 + 17.3i)13-s − 21.8·17-s − 124. i·19-s + (−61.3 − 35.4i)23-s + (52.7 + 91.3i)25-s + (−187. + 108. i)29-s + (242. + 140. i)31-s + (−18.2 − 79.6i)35-s + 150.·37-s + (−136. + 236. i)41-s + (−136. − 236. i)43-s + (−97.3 − 168. i)47-s + ⋯
L(s)  = 1  + (0.197 − 0.341i)5-s + (0.732 − 0.681i)7-s + (1.62 − 0.936i)11-s + (0.639 + 0.369i)13-s − 0.311·17-s − 1.50i·19-s + (−0.556 − 0.321i)23-s + (0.422 + 0.731i)25-s + (−1.20 + 0.694i)29-s + (1.40 + 0.811i)31-s + (−0.0882 − 0.384i)35-s + 0.670·37-s + (−0.519 + 0.900i)41-s + (−0.484 − 0.839i)43-s + (−0.302 − 0.523i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.346 + 0.937i$
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: \((2,\ 756,\ (\ :3/2),\ 0.346 + 0.937i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.578797919\)
\(L(\frac12)\) \(\approx\) \(2.578797919\)
\(L(\frac{5}{2})\) 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 \( 1 + (-13.5 + 12.6i)T \)
good5 \( 1 + (-2.20 + 3.82i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-59.1 + 34.1i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-29.9 - 17.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 21.8T + 4.91e3T^{2} \)
19 \( 1 + 124. iT - 6.85e3T^{2} \)
23 \( 1 + (61.3 + 35.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (187. - 108. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-242. - 140. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 150.T + 5.06e4T^{2} \)
41 \( 1 + (136. - 236. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (136. + 236. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (97.3 + 168. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 520. iT - 1.48e5T^{2} \)
59 \( 1 + (301. - 521. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-145. + 84.0i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-371. + 643. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 758. iT - 3.57e5T^{2} \)
73 \( 1 - 1.15e3iT - 3.89e5T^{2} \)
79 \( 1 + (-78.6 - 136. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-137. - 237. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.04e3T + 7.04e5T^{2} \)
97 \( 1 + (-211. + 122. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−9.634802744837471301683521528876, −8.843502570660455704535128557975, −8.318449860892761276974170524563, −6.97421762598587910831862046631, −6.44592462895345155983156036106, −5.18562718303057286541687168368, −4.28285820429885091996995662680, −3.36111223636476747243438586614, −1.66943332189690130831154062055, −0.78379525824557252522653894292, 1.34507756856279523156559571551, 2.25472518542277646081379763207, 3.74927527130797951405586238916, 4.56407365226595448699438767569, 5.91687325403459979363350676269, 6.38758777109994381648376626015, 7.64955892982106076115295141468, 8.360816318791267501115366442087, 9.352537953280986954216872317323, 9.987701231075955036609263159784

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