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Properties

Label 2-756-63.38-c3-0-14
Degree $2$
Conductor $756$
Sign $0.979 - 0.199i$
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  + (−9.18 + 15.9i)5-s + (18.3 + 2.68i)7-s + (47.9 − 27.6i)11-s + (68.0 − 39.3i)13-s + (−9.99 + 17.3i)17-s + (−16.7 + 9.67i)19-s + (−107. − 62.0i)23-s + (−106. − 184. i)25-s + (48.2 + 27.8i)29-s − 317. i·31-s + (−211. + 266. i)35-s + (20.5 + 35.5i)37-s + (53.1 + 92.0i)41-s + (279. − 483. i)43-s − 147.·47-s + ⋯
L(s)  = 1  + (−0.821 + 1.42i)5-s + (0.989 + 0.145i)7-s + (1.31 − 0.758i)11-s + (1.45 − 0.838i)13-s + (−0.142 + 0.246i)17-s + (−0.202 + 0.116i)19-s + (−0.974 − 0.562i)23-s + (−0.851 − 1.47i)25-s + (0.309 + 0.178i)29-s − 1.84i·31-s + (−1.01 + 1.28i)35-s + (0.0912 + 0.157i)37-s + (0.202 + 0.350i)41-s + (0.990 − 1.71i)43-s − 0.459·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.979 - 0.199i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ 0.979 - 0.199i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.259782261\)
\(L(\frac12)\) \(\approx\) \(2.259782261\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.3 - 2.68i)T \)
good5 \( 1 + (9.18 - 15.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-47.9 + 27.6i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-68.0 + 39.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (9.99 - 17.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (16.7 - 9.67i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (107. + 62.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-48.2 - 27.8i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 317. iT - 2.97e4T^{2} \)
37 \( 1 + (-20.5 - 35.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-53.1 - 92.0i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-279. + 483. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 147.T + 1.03e5T^{2} \)
53 \( 1 + (-570. - 329. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 568.T + 2.05e5T^{2} \)
61 \( 1 - 140. iT - 2.26e5T^{2} \)
67 \( 1 + 294.T + 3.00e5T^{2} \)
71 \( 1 + 602. iT - 3.57e5T^{2} \)
73 \( 1 + (-541. - 312. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 534.T + 4.93e5T^{2} \)
83 \( 1 + (-352. + 610. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-38.6 - 66.9i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-382. - 220. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−10.33348257628155594928325712452, −8.871182419730230729842825694590, −8.222605295922943643565529794910, −7.47615190692435808534505184522, −6.37466190004248726762627880170, −5.83735235357382563212775378135, −4.07997450300186168039630955822, −3.66372399333307708650224839282, −2.34589765743428246478364143400, −0.823023032182387574216262241486, 1.03569333166350303940675399218, 1.68896549790182165035583173783, 3.90401105898442909357245543620, 4.28281126177163447645920546289, 5.21272680172150335521516273153, 6.45984391184953975872316299262, 7.45340475391133359596043289953, 8.493024255646135952840708626891, 8.769052579936611724481192265731, 9.711554617245448733856115124633

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