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Properties

Label 2-756-84.23-c1-0-33
Degree $2$
Conductor $756$
Sign $0.999 - 0.0177i$
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  + (−0.297 + 1.38i)2-s + (−1.82 − 0.822i)4-s + (0.479 + 0.277i)5-s + (−2.45 + 0.976i)7-s + (1.67 − 2.27i)8-s + (−0.525 + 0.580i)10-s + (−2.96 − 5.14i)11-s + 3.20·13-s + (−0.619 − 3.69i)14-s + (2.64 + 2.99i)16-s + (2.48 − 1.43i)17-s + (3.43 + 1.98i)19-s + (−0.646 − 0.899i)20-s + (7.99 − 2.57i)22-s + (0.145 − 0.251i)23-s + ⋯
L(s)  = 1  + (−0.210 + 0.977i)2-s + (−0.911 − 0.411i)4-s + (0.214 + 0.123i)5-s + (−0.929 + 0.369i)7-s + (0.593 − 0.804i)8-s + (−0.166 + 0.183i)10-s + (−0.895 − 1.55i)11-s + 0.888·13-s + (−0.165 − 0.986i)14-s + (0.661 + 0.749i)16-s + (0.602 − 0.348i)17-s + (0.788 + 0.455i)19-s + (−0.144 − 0.201i)20-s + (1.70 − 0.549i)22-s + (0.0303 − 0.0525i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.999 - 0.0177i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.999 - 0.0177i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02060 + 0.00904884i\)
\(L(\frac12)\) \(\approx\) \(1.02060 + 0.00904884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.297 - 1.38i)T \)
3 \( 1 \)
7 \( 1 + (2.45 - 0.976i)T \)
good5 \( 1 + (-0.479 - 0.277i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.96 + 5.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.20T + 13T^{2} \)
17 \( 1 + (-2.48 + 1.43i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.43 - 1.98i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.145 + 0.251i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.13iT - 29T^{2} \)
31 \( 1 + (-5.96 + 3.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.20 + 2.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.27iT - 41T^{2} \)
43 \( 1 - 8.31iT - 43T^{2} \)
47 \( 1 + (-6.19 + 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.21 + 2.43i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.24 + 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.305 - 0.529i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.70 - 2.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.00T + 71T^{2} \)
73 \( 1 + (-6.28 - 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.20 - 3.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.675T + 83T^{2} \)
89 \( 1 + (11.1 + 6.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.00T + 97T^{2} \)
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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

−10.03130858210164760081278087020, −9.492462129910539486739161621458, −8.355919282644365213937384419781, −7.986653529627253134365718006261, −6.67560841915811382175590565793, −5.91845437242180709309694085020, −5.46097084901521189553826479509, −3.90647452850002518633156798483, −2.88591134264598903649276377755, −0.64512991911210722070164366385, 1.27885407741744726417908926682, 2.67231169172223851740595570638, 3.64338933058130102318395238702, 4.71947622708698432952471596304, 5.70673273282425546682248473274, 7.08307480195999071953044746675, 7.80824208285915510918440718513, 8.987364944225895922973468123609, 9.651338291366899505466139337908, 10.33294464314171231033252901938

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