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Properties

Label 2-1456-13.9-c1-0-17
Degree $2$
Conductor $1456$
Sign $0.943 - 0.332i$
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  + (0.682 − 1.18i)3-s + 0.741·5-s + (−0.5 − 0.866i)7-s + (0.568 + 0.984i)9-s + (−0.682 + 1.18i)11-s + (0.301 + 3.59i)13-s + (0.505 − 0.875i)15-s + (2.07 + 3.59i)17-s + (3.63 + 6.29i)19-s − 1.36·21-s + (−1.16 + 2.02i)23-s − 4.45·25-s + 5.64·27-s + (0.203 − 0.353i)29-s + 2.77·31-s + ⋯
L(s)  = 1  + (0.393 − 0.682i)3-s + 0.331·5-s + (−0.188 − 0.327i)7-s + (0.189 + 0.328i)9-s + (−0.205 + 0.356i)11-s + (0.0837 + 0.996i)13-s + (0.130 − 0.226i)15-s + (0.503 + 0.871i)17-s + (0.833 + 1.44i)19-s − 0.297·21-s + (−0.243 + 0.421i)23-s − 0.890·25-s + 1.08·27-s + (0.0378 − 0.0655i)29-s + 0.497·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.943 - 0.332i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.014038770\)
\(L(\frac12)\) \(\approx\) \(2.014038770\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.301 - 3.59i)T \)
good3 \( 1 + (-0.682 + 1.18i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 0.741T + 5T^{2} \)
11 \( 1 + (0.682 - 1.18i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.07 - 3.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.63 - 6.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.16 - 2.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.203 + 0.353i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.77T + 31T^{2} \)
37 \( 1 + (-3.05 + 5.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.627 - 1.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.870 + 1.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.85T + 47T^{2} \)
53 \( 1 - 4.56T + 53T^{2} \)
59 \( 1 + (5.49 + 9.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.26 + 5.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.87 - 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.40 + 4.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.06T + 73T^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + (-0.880 + 1.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.76 + 8.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.728250503174170081742287842246, −8.630332644943881083362716044469, −7.77898505891781434774409746828, −7.36111478184012202532719002678, −6.32493550313363052670740481603, −5.60282152329474840186423339958, −4.38081502537065768309271074760, −3.50150633354913025554777486895, −2.14270466947810786614840309162, −1.43424629329946071499304971988, 0.831880886704817762985827163066, 2.69057071162143407930115354112, 3.20832838586877892113266763240, 4.41146837290628423955201713824, 5.27757422039931692455945085994, 6.07476425103664923898056815232, 7.09222081988616595045726289691, 7.972204278077721406822131961144, 8.879427740325378781534541944255, 9.497476934780960752952107427058

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