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Properties

Label 2-1456-13.3-c1-0-18
Degree $2$
Conductor $1456$
Sign $0.755 + 0.655i$
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  + (−0.366 − 0.633i)3-s − 1.73·5-s + (0.5 − 0.866i)7-s + (1.23 − 2.13i)9-s + (2.36 + 4.09i)11-s + (−1.59 + 3.23i)13-s + (0.633 + 1.09i)15-s + (2.13 − 3.69i)17-s + (1 − 1.73i)19-s − 0.732·21-s + (0.633 + 1.09i)23-s − 2.00·25-s − 4·27-s + (1.5 + 2.59i)29-s + 6.19·31-s + ⋯
L(s)  = 1  + (−0.211 − 0.366i)3-s − 0.774·5-s + (0.188 − 0.327i)7-s + (0.410 − 0.711i)9-s + (0.713 + 1.23i)11-s + (−0.443 + 0.896i)13-s + (0.163 + 0.283i)15-s + (0.517 − 0.896i)17-s + (0.229 − 0.397i)19-s − 0.159·21-s + (0.132 + 0.228i)23-s − 0.400·25-s − 0.769·27-s + (0.278 + 0.482i)29-s + 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.425596659\)
\(L(\frac12)\) \(\approx\) \(1.425596659\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 3.23i)T \)
good3 \( 1 + (0.366 + 0.633i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 + (-2.36 - 4.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.13 + 3.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.633 - 1.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.19T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.59 + 4.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.09 + 8.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.928T + 47T^{2} \)
53 \( 1 - 3.92T + 53T^{2} \)
59 \( 1 + (-5.36 + 9.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.59 + 13.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.09 - 3.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.19T + 73T^{2} \)
79 \( 1 + 5.80T + 79T^{2} \)
83 \( 1 + 8.19T + 83T^{2} \)
89 \( 1 + (-0.464 - 0.803i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.19 + 12.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.616488750645399945227898919704, −8.621537195221152480673771431565, −7.51110433974614563623548979032, −7.08274791519203116948490134743, −6.48896857133585239427079346556, −5.07081372157577007559282844953, −4.32409102646498462230828961903, −3.54042598904400166366086807457, −2.05543919684976084412871943689, −0.795615650694856610750839538500, 1.00596045686711605099685848498, 2.63259739101822689223918417156, 3.73331909517269818993692426136, 4.43064714454983425593232676548, 5.55125554662108330319056916368, 6.10980027143038566025774856880, 7.41042085827055955257982365680, 8.077046526281536191521828981111, 8.584930591121233417729277213058, 9.783954762350956026291861784282

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