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Properties

Label 2-1456-1456.181-c0-0-2
Degree $2$
Conductor $1456$
Sign $0.382 + 0.923i$
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  + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (−1 − i)5-s − 1.00i·6-s + i·7-s + (0.707 − 0.707i)8-s + 1.41i·10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (0.707 − 0.707i)14-s − 1.41i·15-s − 1.00·16-s − 1.41i·17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (−1 − i)5-s − 1.00i·6-s + i·7-s + (0.707 − 0.707i)8-s + 1.41i·10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (0.707 − 0.707i)14-s − 1.41i·15-s − 1.00·16-s − 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7943709995\)
\(L(\frac12)\) \(\approx\) \(0.7943709995\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - iT \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
5 \( 1 + (1 + i)T + iT^{2} \)
11 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
67 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.453487751431853874507940859955, −8.951629220603304396089525305452, −8.136040500683675014771063120012, −7.81143011459270003739699461823, −6.38826322054150295483372118050, −4.97921882862202354884911102178, −4.30595754110760911123040272890, −3.27883140947481867352393567994, −2.66619044954117706558707547660, −0.839711276767404042315579584332, 1.45214382153833589145246417655, 2.62048864587412271575906031805, 3.97518301883579947916515971912, 4.70523839952560193256835874542, 6.48363387934582592404623312465, 6.84569040751196669007112261033, 7.42696404014848385227541317494, 8.094433607243641353199926749438, 8.708221252493152252380756792773, 9.855953613957683883025257627940

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