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Properties

Label 2-1456-13.10-c1-0-11
Degree $2$
Conductor $1456$
Sign $-0.979 - 0.203i$
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  + (1.13 + 1.95i)3-s + 3.60i·5-s + (0.866 + 0.5i)7-s + (−1.05 + 1.83i)9-s + (−0.767 + 0.443i)11-s + (−1.17 − 3.40i)13-s + (−7.05 + 4.07i)15-s + (−2.48 + 4.29i)17-s + (−2.06 − 1.18i)19-s + 2.26i·21-s + (1.92 + 3.34i)23-s − 7.97·25-s + 2.00·27-s + (−0.640 − 1.11i)29-s + 8.46i·31-s + ⋯
L(s)  = 1  + (0.652 + 1.13i)3-s + 1.61i·5-s + (0.327 + 0.188i)7-s + (−0.352 + 0.610i)9-s + (−0.231 + 0.133i)11-s + (−0.325 − 0.945i)13-s + (−1.82 + 1.05i)15-s + (−0.601 + 1.04i)17-s + (−0.472 − 0.272i)19-s + 0.493i·21-s + (0.402 + 0.696i)23-s − 1.59·25-s + 0.385·27-s + (−0.119 − 0.206i)29-s + 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.979 - 0.203i$
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: \((2,\ 1456,\ (\ :1/2),\ -0.979 - 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.803846944\)
\(L(\frac12)\) \(\approx\) \(1.803846944\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 + 3.40i)T \)
good3 \( 1 + (-1.13 - 1.95i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.60iT - 5T^{2} \)
11 \( 1 + (0.767 - 0.443i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.48 - 4.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.06 + 1.18i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.92 - 3.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.640 + 1.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.46iT - 31T^{2} \)
37 \( 1 + (8.34 - 4.81i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.4 + 6.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.82 + 3.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.98iT - 47T^{2} \)
53 \( 1 - 4.92T + 53T^{2} \)
59 \( 1 + (6.34 + 3.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.769 + 1.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.29 - 4.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.58 - 3.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.14iT - 73T^{2} \)
79 \( 1 + 0.757T + 79T^{2} \)
83 \( 1 + 4.76iT - 83T^{2} \)
89 \( 1 + (-3.13 + 1.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.401 + 0.231i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.12308450743825200368800348346, −9.111351684080158859799715221176, −8.436360869501411828322763267743, −7.48713588471463265794813931315, −6.74357015960876714739074820700, −5.72866077646951901839855147800, −4.73341774870123347527044182950, −3.67678361341327965878529477796, −3.09024433589455413155564731862, −2.16388419133479623221894232064, 0.64548028605609587892831055200, 1.73480163256160852770366877175, 2.57228156945854995467007124241, 4.24208053802204274220098094114, 4.78291773825715870629017246488, 5.88272158840973814355902325806, 6.93673629015138771923770270367, 7.65042234319235724482969038815, 8.320203755652059085293342325080, 9.031739706635254122536123930589

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