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Properties

Label 2-1456-364.51-c0-0-1
Degree $2$
Conductor $1456$
Sign $-0.444 + 0.895i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
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  + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 − 1.5i)11-s − 13-s + (−0.5 − 0.866i)17-s + (0.866 − 1.5i)19-s + (−0.5 − 0.866i)25-s + 29-s + (−0.866 + 1.5i)47-s + (0.499 + 0.866i)49-s + (0.5 + 0.866i)53-s + (−0.866 − 1.5i)59-s + (0.5 − 0.866i)61-s + (0.866 − 0.499i)63-s + (0.866 + 1.5i)67-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 − 1.5i)11-s − 13-s + (−0.5 − 0.866i)17-s + (0.866 − 1.5i)19-s + (−0.5 − 0.866i)25-s + 29-s + (−0.866 + 1.5i)47-s + (0.499 + 0.866i)49-s + (0.5 + 0.866i)53-s + (−0.866 − 1.5i)59-s + (0.5 − 0.866i)61-s + (0.866 − 0.499i)63-s + (0.866 + 1.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :0),\ -0.444 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5689256133\)
\(L(\frac12)\) \(\approx\) \(0.5689256133\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.513091631782864405815782165670, −8.630809229785090881061706978573, −7.82389699793931081997643386675, −7.09277611261934428295668939187, −6.17604399985174932540425413082, −5.24148093461228749082455021219, −4.53834147696297994328450051907, −2.95985109187033292458602726178, −2.70782372272853339183490222154, −0.42867665572713238582129033731, 1.94106077312224771018516521648, 2.97711242084772955143775060097, 3.92778438726068339948459157724, 5.10327073456231604566087478709, 5.84866418109046427971054032468, 6.75143734967383911641861972985, 7.49878735662482373853355028797, 8.393253281062428103333980854183, 9.347363700035996941612907318471, 9.940377319554118547616300945720

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