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Properties

Label 2-756-12.11-c1-0-25
Degree $2$
Conductor $756$
Sign $-0.0809 + 0.996i$
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  + (0.0572 − 1.41i)2-s + (−1.99 − 0.161i)4-s + 0.386i·5-s − i·7-s + (−0.342 + 2.80i)8-s + (0.545 + 0.0221i)10-s + 3.49·11-s + 5.33·13-s + (−1.41 − 0.0572i)14-s + (3.94 + 0.645i)16-s − 4.58i·17-s + 6.22i·19-s + (0.0625 − 0.770i)20-s + (0.200 − 4.94i)22-s − 8.20·23-s + ⋯
L(s)  = 1  + (0.0404 − 0.999i)2-s + (−0.996 − 0.0809i)4-s + 0.172i·5-s − 0.377i·7-s + (−0.121 + 0.992i)8-s + (0.172 + 0.00699i)10-s + 1.05·11-s + 1.47·13-s + (−0.377 − 0.0153i)14-s + (0.986 + 0.161i)16-s − 1.11i·17-s + 1.42i·19-s + (0.0139 − 0.172i)20-s + (0.0426 − 1.05i)22-s − 1.71·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0809 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0809 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.0809 + 0.996i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.0809 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02056 - 1.10676i\)
\(L(\frac12)\) \(\approx\) \(1.02056 - 1.10676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0572 + 1.41i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 0.386iT - 5T^{2} \)
11 \( 1 - 3.49T + 11T^{2} \)
13 \( 1 - 5.33T + 13T^{2} \)
17 \( 1 + 4.58iT - 17T^{2} \)
19 \( 1 - 6.22iT - 19T^{2} \)
23 \( 1 + 8.20T + 23T^{2} \)
29 \( 1 + 6.76iT - 29T^{2} \)
31 \( 1 + 9.47iT - 31T^{2} \)
37 \( 1 - 4.41T + 37T^{2} \)
41 \( 1 + 4.43iT - 41T^{2} \)
43 \( 1 + 1.95iT - 43T^{2} \)
47 \( 1 + 0.996T + 47T^{2} \)
53 \( 1 - 1.92iT - 53T^{2} \)
59 \( 1 - 5.78T + 59T^{2} \)
61 \( 1 - 6.68T + 61T^{2} \)
67 \( 1 - 3.49iT - 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 - 8.14T + 73T^{2} \)
79 \( 1 - 12.0iT - 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 0.206iT - 89T^{2} \)
97 \( 1 + 8.99T + 97T^{2} \)
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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

−10.05964343446521738327793677903, −9.578406084885937706172931258471, −8.528041804598448243496957784039, −7.81253881962319642411382918876, −6.41370479401373490221597666337, −5.63641988080407712453118776015, −4.08066788232250234492912686296, −3.78720958926826473044111259186, −2.26811984593341292891808266135, −0.954061671163243584146768346798, 1.31415013070035552493998734878, 3.39226806659464607825819790359, 4.29448261324616215859389691495, 5.36052597054041432355246961150, 6.38039575774449631056139940072, 6.77646443303465172783089658117, 8.202658210433364728162164202765, 8.674139218004920656648724163665, 9.350517703807620721507788995854, 10.46911634628653961957607556217

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