Properties

Label 2-756-63.47-c1-0-3
Degree $2$
Conductor $756$
Sign $0.768 - 0.640i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.91·5-s + (−2.51 + 0.813i)7-s + 3.69i·11-s + (−0.480 + 0.277i)13-s + (2.91 + 5.05i)17-s + (4.62 + 2.66i)19-s − 2.27i·23-s + 10.3·25-s + (−3.53 − 2.04i)29-s + (−7.00 − 4.04i)31-s + (−9.85 + 3.18i)35-s + (3.89 − 6.75i)37-s + (3.59 + 6.22i)41-s + (−0.754 + 1.30i)43-s + (1.41 + 2.44i)47-s + ⋯
L(s)  = 1  + 1.75·5-s + (−0.951 + 0.307i)7-s + 1.11i·11-s + (−0.133 + 0.0769i)13-s + (0.707 + 1.22i)17-s + (1.06 + 0.612i)19-s − 0.474i·23-s + 2.06·25-s + (−0.656 − 0.379i)29-s + (−1.25 − 0.726i)31-s + (−1.66 + 0.538i)35-s + (0.640 − 1.11i)37-s + (0.561 + 0.971i)41-s + (−0.114 + 0.199i)43-s + (0.206 + 0.357i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.768 - 0.640i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.768 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76802 + 0.640281i\)
\(L(\frac12)\) \(\approx\) \(1.76802 + 0.640281i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.51 - 0.813i)T \)
good5 \( 1 - 3.91T + 5T^{2} \)
11 \( 1 - 3.69iT - 11T^{2} \)
13 \( 1 + (0.480 - 0.277i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.91 - 5.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.62 - 2.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.27iT - 23T^{2} \)
29 \( 1 + (3.53 + 2.04i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.00 + 4.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.89 + 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.59 - 6.22i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.754 - 1.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.41 - 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0415 - 0.0239i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.45 + 7.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.03 + 3.48i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.587 - 1.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.71iT - 71T^{2} \)
73 \( 1 + (3.52 - 2.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.97 - 3.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.84 + 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.71 - 4.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.9 + 8.07i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.967890379367030907306191986238, −9.797256005145620912327332166855, −9.107423899108436203817406150303, −7.79243802323234952419037270722, −6.76697582319221026862964510276, −5.89493849579753793977068134661, −5.42418246577792472069614890686, −3.93084765819788158221274574086, −2.59201403300517548513061513233, −1.64555381159723351306148512587, 1.04850725675199135865268219320, 2.63981343261232145048377213781, 3.43613941624963095637040315725, 5.27200696228402773036830995155, 5.65951385506411205025954630253, 6.68407427050467456161649044692, 7.44373786798565931701586367468, 8.976302788497484301224039674957, 9.394043701125595684406459477040, 10.09666039295066296361674168624

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