Properties

Label 2-756-63.41-c3-0-12
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: $\chi_{756} (125, \cdot )$
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

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

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