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Properties

Label 2-1456-1456.1413-c0-0-0
Degree $2$
Conductor $1456$
Sign $-0.921 - 0.388i$
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  + (−0.5 + 0.866i)2-s + (0.965 − 0.258i)3-s + (−0.499 − 0.866i)4-s + (−0.707 + 0.707i)5-s + (−0.258 + 0.965i)6-s + (−0.866 + 0.5i)7-s + 0.999·8-s + (−0.258 − 0.965i)10-s + (−0.707 − 0.707i)12-s + (−0.258 + 0.965i)13-s − 0.999i·14-s + (−0.500 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.965 + 0.258i)20-s + (−0.707 + 0.707i)21-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.965 − 0.258i)3-s + (−0.499 − 0.866i)4-s + (−0.707 + 0.707i)5-s + (−0.258 + 0.965i)6-s + (−0.866 + 0.5i)7-s + 0.999·8-s + (−0.258 − 0.965i)10-s + (−0.707 − 0.707i)12-s + (−0.258 + 0.965i)13-s − 0.999i·14-s + (−0.500 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.965 + 0.258i)20-s + (−0.707 + 0.707i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.921 - 0.388i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (1413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :0),\ -0.921 - 0.388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6231409370\)
\(L(\frac12)\) \(\approx\) \(0.6231409370\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.258 - 0.965i)T \)
good3 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 - 1.41T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.765117842677891122186705367579, −9.000613327450645978647665340544, −8.473255227404949020900930978297, −7.65351297228481624734575856588, −6.94775405892864555611710230710, −6.37159764073773852814817909220, −5.26339975756036807468438310409, −4.02712648392300483031749574975, −3.09248506125046979356690517063, −1.99917101183202042559393808497, 0.51191966608628537758498591436, 2.26678334495370230901696414870, 3.32938365779503436559575671916, 3.80841428105035049665092492889, 4.73644775602777376701980751237, 6.09750792387131589746551924661, 7.53921051118460846435601477860, 7.939740662128963866244821651056, 8.653806113327553358064823870346, 9.492527561368671968172858746919

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