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Properties

Label 2-756-63.20-c3-0-16
Degree $2$
Conductor $756$
Sign $0.320 + 0.947i$
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  + (−10.5 − 18.2i)5-s + (17.6 − 5.46i)7-s + (21.5 + 12.4i)11-s + (52.5 − 30.3i)13-s + 117.·17-s + 104. i·19-s + (17.4 − 10.0i)23-s + (−160. + 277. i)25-s + (24.2 + 14.0i)29-s + (216. − 125. i)31-s + (−286. − 265. i)35-s + 18.2·37-s + (153. + 265. i)41-s + (74.5 − 129. i)43-s + (108. − 188. i)47-s + ⋯
L(s)  = 1  + (−0.944 − 1.63i)5-s + (0.955 − 0.295i)7-s + (0.589 + 0.340i)11-s + (1.12 − 0.647i)13-s + 1.67·17-s + 1.26i·19-s + (0.158 − 0.0915i)23-s + (−1.28 + 2.22i)25-s + (0.155 + 0.0897i)29-s + (1.25 − 0.724i)31-s + (−1.38 − 1.28i)35-s + 0.0809·37-s + (0.583 + 1.00i)41-s + (0.264 − 0.457i)43-s + (0.337 − 0.584i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.320 + 0.947i$
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: \((2,\ 756,\ (\ :3/2),\ 0.320 + 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.300429813\)
\(L(\frac12)\) \(\approx\) \(2.300429813\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.6 + 5.46i)T \)
good5 \( 1 + (10.5 + 18.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-21.5 - 12.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-52.5 + 30.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 117.T + 4.91e3T^{2} \)
19 \( 1 - 104. iT - 6.85e3T^{2} \)
23 \( 1 + (-17.4 + 10.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-24.2 - 14.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-216. + 125. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 18.2T + 5.06e4T^{2} \)
41 \( 1 + (-153. - 265. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-74.5 + 129. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-108. + 188. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 116. iT - 1.48e5T^{2} \)
59 \( 1 + (38.3 + 66.4i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (493. + 285. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (33.8 + 58.6i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 796. iT - 3.57e5T^{2} \)
73 \( 1 - 710. iT - 3.89e5T^{2} \)
79 \( 1 + (40.0 - 69.3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (57.6 - 99.8i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + (444. + 256. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−9.690740896243300994884148699537, −8.705274678710955982018684666239, −7.976866563621285060578916754108, −7.71572919761383681907423567819, −6.02006820067155037167186324615, −5.16465484488047857740969144334, −4.27217164741845711366932605423, −3.55967724897860643521465850422, −1.40685579674606980867393392452, −0.876430950673531330816743497774, 1.09693681597137935926972929516, 2.67717466617202607993608446555, 3.54556868847961890008837586771, 4.47928834929368313002929526265, 5.87407186889374262379048223066, 6.71128763400461356503719384281, 7.52318512233841172868580709154, 8.258217033203639839447998859691, 9.172479829039350315562293250474, 10.45071905211883155082394666273

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