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Properties

Label 2-1456-7.4-c1-0-21
Degree $2$
Conductor $1456$
Sign $0.988 + 0.152i$
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  + (0.879 − 1.52i)3-s + (−0.452 − 0.784i)5-s + (−0.237 + 2.63i)7-s + (−0.0471 − 0.0816i)9-s + (0.358 − 0.620i)11-s + 13-s − 1.59·15-s + (−1.17 + 2.03i)17-s + (3.31 + 5.74i)19-s + (3.80 + 2.67i)21-s + (1.87 + 3.25i)23-s + (2.08 − 3.61i)25-s + 5.11·27-s + 3.25·29-s + (0.785 − 1.36i)31-s + ⋯
L(s)  = 1  + (0.507 − 0.879i)3-s + (−0.202 − 0.350i)5-s + (−0.0898 + 0.995i)7-s + (−0.0157 − 0.0272i)9-s + (0.107 − 0.187i)11-s + 0.277·13-s − 0.411·15-s + (−0.285 + 0.494i)17-s + (0.761 + 1.31i)19-s + (0.830 + 0.584i)21-s + (0.391 + 0.678i)23-s + (0.417 − 0.723i)25-s + 0.983·27-s + 0.604·29-s + (0.141 − 0.244i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.988 + 0.152i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.025310586\)
\(L(\frac12)\) \(\approx\) \(2.025310586\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.237 - 2.63i)T \)
13 \( 1 - T \)
good3 \( 1 + (-0.879 + 1.52i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.452 + 0.784i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.358 + 0.620i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.17 - 2.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.31 - 5.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.87 - 3.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.25T + 29T^{2} \)
31 \( 1 + (-0.785 + 1.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.60 + 4.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.92T + 41T^{2} \)
43 \( 1 - 9.43T + 43T^{2} \)
47 \( 1 + (4.15 + 7.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.04 - 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.358 + 0.620i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.82 - 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.69 + 8.13i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + (-1.73 + 3.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.50 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.54T + 83T^{2} \)
89 \( 1 + (6.02 + 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.43T + 97T^{2} \)
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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

−9.236670978416407231405688104031, −8.572175747946711096424408620441, −7.983957467475380595769794614760, −7.23621478579930500696646571128, −6.19701085291063013359308071023, −5.54635862850692990946959183808, −4.39207735109015201215421249896, −3.22600546699323344128827790776, −2.23806385677537761461986923853, −1.21769271707525856776267933019, 0.927712841614650519625416489895, 2.79128297043353683943189203587, 3.46788370489166103324348031993, 4.43474584542590327585052638079, 5.00303644639597248520019333659, 6.56844367255273669291696568038, 7.02283858039324337824125882210, 7.964589405408792661419896231143, 8.972576143387456445877214197526, 9.501909361088916866424086955479

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