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Properties

Label 2-1456-13.10-c1-0-7
Degree $2$
Conductor $1456$
Sign $-0.166 - 0.986i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (−0.233 − 0.404i)3-s + 3.38i·5-s + (−0.866 − 0.5i)7-s + (1.39 − 2.40i)9-s + (−0.712 + 0.411i)11-s + (−2.74 + 2.33i)13-s + (1.36 − 0.790i)15-s + (2.29 − 3.96i)17-s + (5.11 + 2.95i)19-s + 0.466i·21-s + (3.06 + 5.30i)23-s − 6.48·25-s − 2.69·27-s + (3.43 + 5.94i)29-s − 4.28i·31-s + ⋯
L(s)  = 1  + (−0.134 − 0.233i)3-s + 1.51i·5-s + (−0.327 − 0.188i)7-s + (0.463 − 0.803i)9-s + (−0.214 + 0.124i)11-s + (−0.762 + 0.646i)13-s + (0.353 − 0.204i)15-s + (0.555 − 0.962i)17-s + (1.17 + 0.677i)19-s + 0.101i·21-s + (0.639 + 1.10i)23-s − 1.29·25-s − 0.519·27-s + (0.637 + 1.10i)29-s − 0.769i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.166 - 0.986i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.166 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.252847203\)
\(L(\frac12)\) \(\approx\) \(1.252847203\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (2.74 - 2.33i)T \)
good3 \( 1 + (0.233 + 0.404i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.38iT - 5T^{2} \)
11 \( 1 + (0.712 - 0.411i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.29 + 3.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.11 - 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.06 - 5.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.43 - 5.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.28iT - 31T^{2} \)
37 \( 1 + (8.39 - 4.84i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0774 - 0.0446i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.67 - 6.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 + 7.01T + 53T^{2} \)
59 \( 1 + (1.50 + 0.870i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.18 - 2.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.252 - 0.145i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.48 + 5.47i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + 9.95T + 79T^{2} \)
83 \( 1 - 3.23iT - 83T^{2} \)
89 \( 1 + (-6.96 + 4.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.7 - 7.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.767926810064748678636295696781, −9.278665955233063691589596572101, −7.73014308896816021026003350116, −7.19218826397633976584554408447, −6.73271022394634250602348529936, −5.80415361305540655345277073101, −4.71334702548807426367480897450, −3.35263290015639653000815228832, −2.99460432156263201039063361805, −1.42943910489075793001542364345, 0.53205273654997180018877179389, 1.88685780741591662331148866059, 3.21879954887031448351648823412, 4.47615554606753166330194031995, 5.10685217033616064892985241206, 5.63622658087333474644887881766, 6.96052924454060142229959116101, 7.85210061474650288242646265604, 8.537845054655328624422190205171, 9.222188529610901205198372274554

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