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Properties

Label 2-756-63.20-c3-0-15
Degree $2$
Conductor $756$
Sign $0.143 + 0.989i$
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  + (−5.49 − 9.52i)5-s + (6.85 + 17.2i)7-s + (−13.8 − 7.98i)11-s + (−77.4 + 44.7i)13-s + 106.·17-s + 8.31i·19-s + (123. − 71.5i)23-s + (2.06 − 3.58i)25-s + (−129. − 74.8i)29-s + (−37.1 + 21.4i)31-s + (126. − 159. i)35-s + 390.·37-s + (−172. − 298. i)41-s + (28.5 − 49.4i)43-s + (−8.13 + 14.0i)47-s + ⋯
L(s)  = 1  + (−0.491 − 0.851i)5-s + (0.370 + 0.928i)7-s + (−0.378 − 0.218i)11-s + (−1.65 + 0.953i)13-s + 1.52·17-s + 0.100i·19-s + (1.12 − 0.648i)23-s + (0.0165 − 0.0286i)25-s + (−0.830 − 0.479i)29-s + (−0.215 + 0.124i)31-s + (0.609 − 0.772i)35-s + 1.73·37-s + (−0.657 − 1.13i)41-s + (0.101 − 0.175i)43-s + (−0.0252 + 0.0437i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.143 + 0.989i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ 0.143 + 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.297192640\)
\(L(\frac12)\) \(\approx\) \(1.297192640\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.85 - 17.2i)T \)
good5 \( 1 + (5.49 + 9.52i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (13.8 + 7.98i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (77.4 - 44.7i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 106.T + 4.91e3T^{2} \)
19 \( 1 - 8.31iT - 6.85e3T^{2} \)
23 \( 1 + (-123. + 71.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (129. + 74.8i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (37.1 - 21.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 390.T + 5.06e4T^{2} \)
41 \( 1 + (172. + 298. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-28.5 + 49.4i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (8.13 - 14.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 445. iT - 1.48e5T^{2} \)
59 \( 1 + (193. + 334. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (420. + 242. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (251. + 436. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 751. iT - 3.57e5T^{2} \)
73 \( 1 + 507. iT - 3.89e5T^{2} \)
79 \( 1 + (-381. + 660. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-607. + 1.05e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 425.T + 7.04e5T^{2} \)
97 \( 1 + (-494. - 285. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−9.478102114534421172559302032025, −9.031260357228243162692775867821, −7.967823505080250185455303717856, −7.43483515882046322647574251853, −6.06640161142353361321588463099, −5.07210281997027963480609630381, −4.55059134172288191493394643649, −3.06006330145209946708907416766, −1.91609411681371066622705774915, −0.41538388264360298633533894919, 1.02723832819527432230039821973, 2.73084475150361620076634219117, 3.49651646874149505063786851531, 4.76466812681626726712253737712, 5.56085839532045419782093625413, 7.06840218647774164711129611130, 7.46167362060802749587962371050, 8.054247279052588576325349095406, 9.632900203614562816590999305482, 10.11067888109808115973704986250

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