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Properties

Label 2-756-63.16-c1-0-6
Degree $2$
Conductor $756$
Sign $0.858 + 0.512i$
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  + 2.52·5-s + (1.98 − 1.75i)7-s + 1.37·11-s + (−2.80 − 4.84i)13-s + (2.69 + 4.66i)17-s + (2.44 − 4.23i)19-s − 4.17·23-s + 1.35·25-s + (1.56 − 2.71i)29-s + (−2.40 + 4.15i)31-s + (4.99 − 4.41i)35-s + (−2.69 + 4.67i)37-s + (3.02 + 5.24i)41-s + (2.44 − 4.23i)43-s + (−2.82 − 4.89i)47-s + ⋯
L(s)  = 1  + 1.12·5-s + (0.748 − 0.662i)7-s + 0.414·11-s + (−0.776 − 1.34i)13-s + (0.653 + 1.13i)17-s + (0.561 − 0.972i)19-s − 0.870·23-s + 0.270·25-s + (0.291 − 0.504i)29-s + (−0.431 + 0.746i)31-s + (0.844 − 0.746i)35-s + (−0.443 + 0.768i)37-s + (0.473 + 0.819i)41-s + (0.373 − 0.646i)43-s + (−0.412 − 0.714i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.858 + 0.512i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.858 + 0.512i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92445 - 0.530844i\)
\(L(\frac12)\) \(\approx\) \(1.92445 - 0.530844i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.98 + 1.75i)T \)
good5 \( 1 - 2.52T + 5T^{2} \)
11 \( 1 - 1.37T + 11T^{2} \)
13 \( 1 + (2.80 + 4.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.69 - 4.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.44 + 4.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.17T + 23T^{2} \)
29 \( 1 + (-1.56 + 2.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.40 - 4.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.69 - 4.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.02 - 5.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.44 + 4.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.82 + 4.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.00 - 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.13 + 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.42 - 5.93i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.05 + 7.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.25T + 71T^{2} \)
73 \( 1 + (-3.51 - 6.08i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.48 - 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.75 - 4.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.894 + 1.54i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.16891390297477467827542190704, −9.695337894295753016743547553957, −8.473951154494216590980198240903, −7.74855330055350659366438100398, −6.76339349352220281382513235898, −5.69509246917029091289680917052, −5.04639448020316402994049832325, −3.77855932062066920584113410723, −2.43177039829684986986307852413, −1.17799676900978505759666949833, 1.64360433446543861869307594038, 2.46397041551868884092406201918, 4.06514467729176962170605032708, 5.24043541837871563442697978173, 5.80636701768088653983692106597, 6.91645744791344745475030528433, 7.80847090813175762447151160605, 8.957693829754736007115445709378, 9.545819384701114093415205209702, 10.16002394356103706376524360115

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