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Properties

Label 2-1456-13.4-c1-0-23
Degree $2$
Conductor $1456$
Sign $0.991 - 0.129i$
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  + (−1.41 + 2.44i)3-s + 0.518i·5-s + (0.866 − 0.5i)7-s + (−2.49 − 4.31i)9-s + (−1.40 − 0.812i)11-s + (1.42 − 3.31i)13-s + (−1.26 − 0.733i)15-s + (0.974 + 1.68i)17-s + (−2.15 + 1.24i)19-s + 2.82i·21-s + (4.57 − 7.91i)23-s + 4.73·25-s + 5.60·27-s + (2.61 − 4.52i)29-s − 5.79i·31-s + ⋯
L(s)  = 1  + (−0.815 + 1.41i)3-s + 0.232i·5-s + (0.327 − 0.188i)7-s + (−0.830 − 1.43i)9-s + (−0.424 − 0.244i)11-s + (0.395 − 0.918i)13-s + (−0.327 − 0.189i)15-s + (0.236 + 0.409i)17-s + (−0.494 + 0.285i)19-s + 0.616i·21-s + (0.952 − 1.65i)23-s + 0.946·25-s + 1.07·27-s + (0.485 − 0.841i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.991 - 0.129i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.108428418\)
\(L(\frac12)\) \(\approx\) \(1.108428418\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 + 3.31i)T \)
good3 \( 1 + (1.41 - 2.44i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 0.518iT - 5T^{2} \)
11 \( 1 + (1.40 + 0.812i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.974 - 1.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.15 - 1.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.57 + 7.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.61 + 4.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.79iT - 31T^{2} \)
37 \( 1 + (8.85 + 5.11i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.64 - 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.498 - 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.51iT - 47T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 + (-5.37 + 3.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.73 - 11.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.25 - 4.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.50 + 2.59i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + 0.982T + 79T^{2} \)
83 \( 1 - 8.91iT - 83T^{2} \)
89 \( 1 + (10.4 + 6.00i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.82 + 2.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.868937880017665334029250255849, −8.726753687971771773697726389307, −8.195589950756270725176142087818, −6.93024114851158345461845494059, −5.99298022452267472988392814159, −5.32776171473176009808740325637, −4.50939273781209398515638121034, −3.73162594598680162853623328115, −2.65182442789504726519732955685, −0.59144387401092932543557465484, 1.10909008705558905456468872850, 1.92101656813415784775377751416, 3.25344095826931815841350051146, 4.89882774148071598088080197941, 5.29618827925500518521500842648, 6.48486873919834450845105602293, 6.94311845518354095331771373886, 7.69443384801146911673875131384, 8.603131985876215426754583693027, 9.307434418417840798766633412083

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