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Properties

Label 2-1456-13.4-c1-0-25
Degree $2$
Conductor $1456$
Sign $0.925 + 0.378i$
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  + (−0.432 + 0.749i)3-s − 3.71i·5-s + (0.866 − 0.5i)7-s + (1.12 + 1.94i)9-s + (5.00 + 2.88i)11-s + (2.87 − 2.17i)13-s + (2.78 + 1.60i)15-s + (−0.106 − 0.183i)17-s + (−1.85 + 1.06i)19-s + 0.865i·21-s + (−1.23 + 2.14i)23-s − 8.77·25-s − 4.54·27-s + (0.0492 − 0.0853i)29-s − 2.31i·31-s + ⋯
L(s)  = 1  + (−0.249 + 0.432i)3-s − 1.65i·5-s + (0.327 − 0.188i)7-s + (0.375 + 0.649i)9-s + (1.50 + 0.870i)11-s + (0.798 − 0.602i)13-s + (0.718 + 0.414i)15-s + (−0.0257 − 0.0445i)17-s + (−0.424 + 0.245i)19-s + 0.188i·21-s + (−0.258 + 0.447i)23-s − 1.75·25-s − 0.874·27-s + (0.00915 − 0.0158i)29-s − 0.415i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.925 + 0.378i$
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: \((2,\ 1456,\ (\ :1/2),\ 0.925 + 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.867533366\)
\(L(\frac12)\) \(\approx\) \(1.867533366\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.87 + 2.17i)T \)
good3 \( 1 + (0.432 - 0.749i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.71iT - 5T^{2} \)
11 \( 1 + (-5.00 - 2.88i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.106 + 0.183i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.85 - 1.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.23 - 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0492 + 0.0853i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.31iT - 31T^{2} \)
37 \( 1 + (-6.81 - 3.93i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.51 - 3.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.28 + 3.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.15iT - 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + (-0.200 + 0.115i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.01 + 6.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.2 - 6.48i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.37 - 3.68i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.60iT - 73T^{2} \)
79 \( 1 - 9.19T + 79T^{2} \)
83 \( 1 + 3.17iT - 83T^{2} \)
89 \( 1 + (10.2 + 5.90i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.1 + 7.04i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.548267926922611291619803595238, −8.640539672766704629737831397164, −8.096964702003920491014894486994, −7.11108159897606467920096084889, −5.94806046932613990876268429724, −5.18952354739459949557243155286, −4.31016104306448015165316430367, −3.95966361330150216966775110229, −1.90437479559235174283476166947, −1.02798774099105228870049122119, 1.14856827974225212708453970332, 2.42877665874307216410216468317, 3.59564000248060542903832543364, 4.16238596924129882905859566003, 5.96576302939275579835663207341, 6.35738197673128595517247929919, 6.88703028292081633364939471312, 7.78499731738414987915880107967, 8.887912457223201057470213039950, 9.460359639912314773675128274557

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