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Properties

Label 2-1456-13.10-c1-0-35
Degree $2$
Conductor $1456$
Sign $-0.515 + 0.856i$
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  + (−0.172 − 0.299i)3-s − 3.25i·5-s + (−0.866 − 0.5i)7-s + (1.44 − 2.49i)9-s + (1.59 − 0.923i)11-s + (3.60 + 0.0186i)13-s + (−0.976 + 0.563i)15-s + (1.07 − 1.86i)17-s + (2.07 + 1.20i)19-s + 0.345i·21-s + (−0.906 − 1.56i)23-s − 5.61·25-s − 2.03·27-s + (1.36 + 2.36i)29-s + 1.74i·31-s + ⋯
L(s)  = 1  + (−0.0998 − 0.172i)3-s − 1.45i·5-s + (−0.327 − 0.188i)7-s + (0.480 − 0.831i)9-s + (0.482 − 0.278i)11-s + (0.999 + 0.00517i)13-s + (−0.252 + 0.145i)15-s + (0.261 − 0.452i)17-s + (0.477 + 0.275i)19-s + 0.0754i·21-s + (−0.188 − 0.327i)23-s − 1.12·25-s − 0.391·27-s + (0.253 + 0.439i)29-s + 0.312i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.515 + 0.856i$
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: \((2,\ 1456,\ (\ :1/2),\ -0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.628524655\)
\(L(\frac12)\) \(\approx\) \(1.628524655\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 - 0.0186i)T \)
good3 \( 1 + (0.172 + 0.299i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.25iT - 5T^{2} \)
11 \( 1 + (-1.59 + 0.923i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.07 + 1.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.07 - 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.906 + 1.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.36 - 2.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.74iT - 31T^{2} \)
37 \( 1 + (5.14 - 2.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.65 + 2.11i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.34 + 7.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.87iT - 47T^{2} \)
53 \( 1 + 9.30T + 53T^{2} \)
59 \( 1 + (9.31 + 5.37i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.05 - 8.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.716 - 0.413i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.03 - 1.17i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.19iT - 73T^{2} \)
79 \( 1 + 0.801T + 79T^{2} \)
83 \( 1 - 9.97iT - 83T^{2} \)
89 \( 1 + (-13.0 + 7.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.99 - 4.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.064952586667220026413846472648, −8.708536063377199424276847208775, −7.69627791516216038962043455078, −6.72639639884461768334524519954, −5.95889351241621962377492994045, −5.07546466824908946181943924596, −4.09882668280598411389970187596, −3.35701235556704727663728326025, −1.51636491884816718841367731273, −0.72627010920897608189515307983, 1.66373828866415446556220859374, 2.85447271074203300810404726511, 3.67671129253672049903397681426, 4.66587012023391038480896841361, 5.96725891964469530534128140395, 6.41361269240664431385404339238, 7.41866581046447873835485764304, 7.935596409762936953379504803252, 9.194476839628690362017354577834, 9.839946507408284961642611879826

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