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Properties

Label 2-728-728.237-c0-0-3
Degree $2$
Conductor $728$
Sign $0.967 + 0.252i$
Analytic cond. $0.363319$
Root an. cond. $0.602759$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·5-s + (0.866 − 1.5i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−1 + 1.73i)9-s + (0.866 + 1.49i)10-s + 1.73·12-s + (−0.866 − 0.5i)13-s + 0.999·14-s + (−1.49 − 2.59i)15-s + (−0.5 − 0.866i)16-s − 2·18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·5-s + (0.866 − 1.5i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−1 + 1.73i)9-s + (0.866 + 1.49i)10-s + 1.73·12-s + (−0.866 − 0.5i)13-s + 0.999·14-s + (−1.49 − 2.59i)15-s + (−0.5 − 0.866i)16-s − 2·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(0.363319\)
Root analytic conductor: \(0.602759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :0),\ 0.967 + 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.135094935\)
\(L(\frac12)\) \(\approx\) \(1.135094935\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - 1.73T + T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.63157422720753413224646223117, −9.704823732937147800103367237743, −8.548102689221723831599473891750, −7.37503432491491403919223077770, −7.13135071590034197549836871651, −6.04670808622422837535819918405, −5.57357652766491961380423211411, −4.72196742669697703902744266762, −2.70531240019805837894332022205, −1.40302546205491119264690240160, 1.94224099105061124587509288696, 3.01380030288860445188221829542, 4.59982418802210044406978793320, 5.02074188124954739642311961318, 5.79829003412315497443300531174, 6.44955491513201741717966213335, 8.774809372988672521730911430359, 9.414234912420366850561707505944, 9.871378168251772001822184329977, 10.63868279463485881251013491010

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