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Properties

Label 2-756-63.20-c3-0-2
Degree $2$
Conductor $756$
Sign $-0.108 - 0.994i$
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  + (−3.53 − 6.12i)5-s + (10.8 − 15.0i)7-s + (−7.40 − 4.27i)11-s + (−45.3 + 26.1i)13-s − 38.9·17-s + 66.4i·19-s + (−173. + 100. i)23-s + (37.5 − 64.9i)25-s + (−52.9 − 30.5i)29-s + (116. − 67.2i)31-s + (−130. − 13.2i)35-s + 298.·37-s + (221. + 383. i)41-s + (26.1 − 45.2i)43-s + (−137. + 238. i)47-s + ⋯
L(s)  = 1  + (−0.316 − 0.547i)5-s + (0.584 − 0.811i)7-s + (−0.202 − 0.117i)11-s + (−0.967 + 0.558i)13-s − 0.555·17-s + 0.801i·19-s + (−1.57 + 0.910i)23-s + (0.300 − 0.519i)25-s + (−0.339 − 0.195i)29-s + (0.674 − 0.389i)31-s + (−0.629 − 0.0637i)35-s + 1.32·37-s + (0.842 + 1.45i)41-s + (0.0926 − 0.160i)43-s + (−0.427 + 0.740i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.108 - 0.994i$
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: \((2,\ 756,\ (\ :3/2),\ -0.108 - 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7715608289\)
\(L(\frac12)\) \(\approx\) \(0.7715608289\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.8 + 15.0i)T \)
good5 \( 1 + (3.53 + 6.12i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (7.40 + 4.27i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (45.3 - 26.1i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 38.9T + 4.91e3T^{2} \)
19 \( 1 - 66.4iT - 6.85e3T^{2} \)
23 \( 1 + (173. - 100. i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (52.9 + 30.5i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-116. + 67.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 298.T + 5.06e4T^{2} \)
41 \( 1 + (-221. - 383. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-26.1 + 45.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (137. - 238. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 136. iT - 1.48e5T^{2} \)
59 \( 1 + (191. + 331. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-261. - 151. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-318. - 552. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 228. iT - 3.57e5T^{2} \)
73 \( 1 - 1.24e3iT - 3.89e5T^{2} \)
79 \( 1 + (100. - 174. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (323. - 560. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 826.T + 7.04e5T^{2} \)
97 \( 1 + (17.0 + 9.85i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−10.01706588368670490804681380717, −9.508795007847527516259605529336, −8.116598541729940502054190586695, −7.893103807741929450968024280393, −6.76417675084480066327070953478, −5.68258332047995297958678591691, −4.50684796879879519696843062875, −4.06314779714985638454968043193, −2.42299831409341232835666454984, −1.15861838796747705309875477393, 0.21981463436946615282340343921, 2.11288130581262755631612252513, 2.88411359696454882175159736371, 4.31128670546478764021631440762, 5.18360172631328141484980923020, 6.18411973599929973116528151453, 7.21067087218904670964113106198, 7.970602833055024252778953206092, 8.810315582975713464768122209078, 9.747366887358095687408978350485

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