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Properties

Label 2-756-21.17-c3-0-21
Degree $2$
Conductor $756$
Sign $0.932 + 0.361i$
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  + (6.90 + 11.9i)5-s + (8.82 − 16.2i)7-s + (−35.5 − 20.5i)11-s + 57.5i·13-s + (56.4 − 97.7i)17-s + (77.1 − 44.5i)19-s + (−56.2 + 32.5i)23-s + (−32.7 + 56.8i)25-s − 277. i·29-s + (228. + 131. i)31-s + (255. − 6.94i)35-s + (−144. − 251. i)37-s − 140.·41-s + 260.·43-s + (238. + 412. i)47-s + ⋯
L(s)  = 1  + (0.617 + 1.06i)5-s + (0.476 − 0.879i)7-s + (−0.975 − 0.563i)11-s + 1.22i·13-s + (0.804 − 1.39i)17-s + (0.931 − 0.537i)19-s + (−0.510 + 0.294i)23-s + (−0.262 + 0.454i)25-s − 1.77i·29-s + (1.32 + 0.762i)31-s + (1.23 − 0.0335i)35-s + (−0.643 − 1.11i)37-s − 0.536·41-s + 0.923·43-s + (0.739 + 1.28i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.932 + 0.361i$
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: \((2,\ 756,\ (\ :3/2),\ 0.932 + 0.361i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.328188733\)
\(L(\frac12)\) \(\approx\) \(2.328188733\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.82 + 16.2i)T \)
good5 \( 1 + (-6.90 - 11.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (35.5 + 20.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 57.5iT - 2.19e3T^{2} \)
17 \( 1 + (-56.4 + 97.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-77.1 + 44.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (56.2 - 32.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 277. iT - 2.43e4T^{2} \)
31 \( 1 + (-228. - 131. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (144. + 251. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 140.T + 6.89e4T^{2} \)
43 \( 1 - 260.T + 7.95e4T^{2} \)
47 \( 1 + (-238. - 412. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (86.0 + 49.6i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (335. - 580. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-119. + 68.7i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-101. + 176. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 601. iT - 3.57e5T^{2} \)
73 \( 1 + (-832. - 480. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-320. - 554. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.37e3T + 5.71e5T^{2} \)
89 \( 1 + (-181. - 314. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 828. iT - 9.12e5T^{2} \)
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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

−9.963550113993133671813975610250, −9.290958352757350043825900718486, −7.908081943014079864345263491439, −7.32907622235333654835162479198, −6.48468911671773593363684451836, −5.46864733233066531868249746341, −4.44780185632611961438389783535, −3.16951109190336427357388750837, −2.28168344070002638234122909250, −0.74324863002087904343055989322, 1.07897875627408515459387226148, 2.11681046849539970798280743709, 3.38926610450531890032392606638, 5.01742984064690858211260919101, 5.30602719766159099985664238718, 6.18965877925487762140510361883, 7.81420927189559758256478286561, 8.202613545189570615969816670089, 9.085020163449390515499986304181, 10.09855236111329943997354547565

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