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Properties

Label 2-756-63.20-c3-0-1
Degree $2$
Conductor $756$
Sign $-0.968 + 0.247i$
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 + (−4.11 − 18.0i)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 + (−0.222 − 0.974i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.968 + 0.247i$
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: \((2,\ 756,\ (\ :3/2),\ -0.968 + 0.247i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4267284814\)
\(L(\frac12)\) \(\approx\) \(0.4267284814\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.11 + 18.0i)T \)
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

−10.59629628325185188996290381430, −9.559766288354146878299779924443, −9.075521806371047082549437456016, −7.28774854399356749508095426525, −6.94719665906050304314954918948, −6.41609510698681617151346469025, −4.98894248046655906262565761498, −3.92066293547543950787921927786, −2.72519903372378378393290934345, −1.88926746281555011038452120785, 0.10317460867887725741266075875, 1.57273009320458801845526913670, 2.47975484161549947295497160408, 4.12585528743541069005284478219, 5.16600872331252453855084414323, 5.70521497268606395933325446795, 6.63288779180553832830639023356, 8.131888734202531220011723561454, 8.702145751673834665528581325558, 9.495743033496361101410964892086

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