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Properties

Label 2-756-84.11-c1-0-56
Degree $2$
Conductor $756$
Sign $0.126 + 0.991i$
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  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (2.44 − 1.41i)5-s + (−2 + 1.73i)7-s − 2.82i·8-s + (1.99 − 3.46i)10-s + (1.22 − 2.12i)11-s − 13-s + (−1.22 + 3.53i)14-s + (−2.00 − 3.46i)16-s + (1.22 + 0.707i)17-s + (6 − 3.46i)19-s − 5.65i·20-s − 3.46i·22-s + (−1.22 − 2.12i)23-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s + (1.09 − 0.632i)5-s + (−0.755 + 0.654i)7-s − 0.999i·8-s + (0.632 − 1.09i)10-s + (0.369 − 0.639i)11-s − 0.277·13-s + (−0.327 + 0.944i)14-s + (−0.500 − 0.866i)16-s + (0.297 + 0.171i)17-s + (1.37 − 0.794i)19-s − 1.26i·20-s − 0.738i·22-s + (−0.255 − 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.126 + 0.991i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.126 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12582 - 1.87211i\)
\(L(\frac12)\) \(\approx\) \(2.12582 - 1.87211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + (-2.44 + 1.41i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.22 + 2.12i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-1.22 - 0.707i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 + 3.46i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.22 + 2.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.07iT - 29T^{2} \)
31 \( 1 + (4.5 + 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 - 8.66iT - 43T^{2} \)
47 \( 1 + (-3.67 - 6.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.57 - 4.94i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.67 + 6.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.5 - 7.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
89 \( 1 + (12.2 - 7.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5T + 97T^{2} \)
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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

−10.07428467003756580110996844770, −9.418732238419947025473509594300, −8.880455176161535410055861888196, −7.26850089908115807630898782317, −6.19474349965050336560396783214, −5.63561579474437703779566111735, −4.86591354183775118209374626594, −3.45246634867849467478988326928, −2.53702257141894537408450470620, −1.21518807804577845651613813185, 1.98313523846912663254530051732, 3.18413435071329298028850017681, 4.06357833971198367598073755247, 5.44116870148766207212109602560, 6.00150021174281633718225225959, 7.13107352604769912922872734538, 7.33954030476246801607655133222, 8.851398106215484157173243918939, 10.03066297089865919234564227397, 10.19755580651408825078744764093

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