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Properties

Label 2-756-63.20-c3-0-6
Degree $2$
Conductor $756$
Sign $-0.875 - 0.484i$
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  + (7.82 + 13.5i)5-s + (17.6 + 5.69i)7-s + (−34.2 − 19.7i)11-s + (−55.5 + 32.0i)13-s + 56.6·17-s + 117. i·19-s + (6.59 − 3.80i)23-s + (−59.9 + 103. i)25-s + (−39.8 − 22.9i)29-s + (−251. + 145. i)31-s + (60.6 + 283. i)35-s − 335.·37-s + (97.2 + 168. i)41-s + (152. − 263. i)43-s + (318. − 550. i)47-s + ⋯
L(s)  = 1  + (0.699 + 1.21i)5-s + (0.951 + 0.307i)7-s + (−0.937 − 0.541i)11-s + (−1.18 + 0.684i)13-s + 0.808·17-s + 1.41i·19-s + (0.0597 − 0.0345i)23-s + (−0.479 + 0.830i)25-s + (−0.254 − 0.147i)29-s + (−1.45 + 0.842i)31-s + (0.293 + 1.36i)35-s − 1.49·37-s + (0.370 + 0.641i)41-s + (0.540 − 0.935i)43-s + (0.987 − 1.70i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.484i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.875 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.875 - 0.484i$
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: \((2,\ 756,\ (\ :3/2),\ -0.875 - 0.484i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.497672887\)
\(L(\frac12)\) \(\approx\) \(1.497672887\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.6 - 5.69i)T \)
good5 \( 1 + (-7.82 - 13.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (34.2 + 19.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (55.5 - 32.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 56.6T + 4.91e3T^{2} \)
19 \( 1 - 117. iT - 6.85e3T^{2} \)
23 \( 1 + (-6.59 + 3.80i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (39.8 + 22.9i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (251. - 145. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 335.T + 5.06e4T^{2} \)
41 \( 1 + (-97.2 - 168. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-152. + 263. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-318. + 550. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 274. iT - 1.48e5T^{2} \)
59 \( 1 + (258. + 448. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (142. + 82.1i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-368. - 637. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 599. iT - 3.57e5T^{2} \)
73 \( 1 - 214. iT - 3.89e5T^{2} \)
79 \( 1 + (454. - 786. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (389. - 675. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 443.T + 7.04e5T^{2} \)
97 \( 1 + (337. + 194. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−10.43148973775599637421990647801, −9.637640049564871734974973306925, −8.540871872208536649719695185390, −7.60353476375223713822734427574, −6.95744551477443073278785303520, −5.69264420641277689835983806553, −5.24341929504268268480196236638, −3.71314200620863857902123488044, −2.56905775309427243391738397255, −1.75114864595254755776463202619, 0.37378231156413680275352991669, 1.61439638631621045534823297489, 2.67848749290736371881009233754, 4.41582529313076725896878560457, 5.14947848279315953665097388432, 5.59100303759791559225715667366, 7.32705578159394701522439492333, 7.73806526532204078717779910596, 8.852892675451359623704419931471, 9.519013131809707159745056522400

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