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Properties

Label 2-756-21.17-c3-0-12
Degree $2$
Conductor $756$
Sign $-0.210 - 0.977i$
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  + (5.32 + 9.23i)5-s + (18.4 − 1.58i)7-s + (58.6 + 33.8i)11-s + 59.7i·13-s + (−43.3 + 75.1i)17-s + (−125. + 72.1i)19-s + (−5.66 + 3.27i)23-s + (5.70 − 9.87i)25-s − 291. i·29-s + (45.4 + 26.2i)31-s + (112. + 161. i)35-s + (−45.3 − 78.6i)37-s + 188.·41-s + 61.9·43-s + (−24.8 − 43.0i)47-s + ⋯
L(s)  = 1  + (0.476 + 0.825i)5-s + (0.996 − 0.0856i)7-s + (1.60 + 0.928i)11-s + 1.27i·13-s + (−0.618 + 1.07i)17-s + (−1.50 + 0.871i)19-s + (−0.0513 + 0.0296i)23-s + (0.0456 − 0.0790i)25-s − 1.86i·29-s + (0.263 + 0.152i)31-s + (0.545 + 0.781i)35-s + (−0.201 − 0.349i)37-s + 0.716·41-s + 0.219·43-s + (−0.0771 − 0.133i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.210 - 0.977i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ -0.210 - 0.977i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.443745412\)
\(L(\frac12)\) \(\approx\) \(2.443745412\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 + 1.58i)T \)
good5 \( 1 + (-5.32 - 9.23i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-58.6 - 33.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 59.7iT - 2.19e3T^{2} \)
17 \( 1 + (43.3 - 75.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (125. - 72.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (5.66 - 3.27i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 291. iT - 2.43e4T^{2} \)
31 \( 1 + (-45.4 - 26.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (45.3 + 78.6i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 188.T + 6.89e4T^{2} \)
43 \( 1 - 61.9T + 7.95e4T^{2} \)
47 \( 1 + (24.8 + 43.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (267. + 154. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (313. - 542. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (534. - 308. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (78.5 - 136. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 915. iT - 3.57e5T^{2} \)
73 \( 1 + (105. + 60.6i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-345. - 598. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 16.1T + 5.71e5T^{2} \)
89 \( 1 + (492. + 852. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.69e3iT - 9.12e5T^{2} \)
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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

−10.27580591873596833723226847729, −9.334474943773428578894587024091, −8.568881302277071485143905223477, −7.52934963421597372716142508764, −6.45566074119412527538818573678, −6.19757188557729448180744472250, −4.31938959155186412401470276485, −4.15484068871231210485519882682, −2.16693809702392457370895109718, −1.65380973090657511994811013814, 0.66569881824435156939133485387, 1.62177384401196561571927199751, 3.04341035420721709033895831459, 4.42011128709515626366852077050, 5.11825915535769413483761609213, 6.08998877924381528436475790853, 7.04235275512864171137272268946, 8.284025832929036479802206090450, 8.847505383287421681002716994976, 9.408787001115777501320538951440

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