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Properties

Label 2-2548-2548.571-c0-0-1
Degree $2$
Conductor $2548$
Sign $0.977 - 0.212i$
Analytic cond. $1.27161$
Root an. cond. $1.12766$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.988 − 0.149i)7-s + (−0.222 − 0.974i)8-s + (0.826 + 0.563i)9-s + (1.57 − 1.07i)11-s + (−0.900 − 0.433i)13-s + (0.0747 − 0.997i)14-s + (0.955 − 0.294i)16-s + (0.0546 + 0.139i)17-s + (−0.5 + 0.866i)18-s + (−0.826 − 1.43i)19-s + (1.19 + 1.49i)22-s + (0.0747 − 0.997i)25-s + (0.365 − 0.930i)26-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.988 − 0.149i)7-s + (−0.222 − 0.974i)8-s + (0.826 + 0.563i)9-s + (1.57 − 1.07i)11-s + (−0.900 − 0.433i)13-s + (0.0747 − 0.997i)14-s + (0.955 − 0.294i)16-s + (0.0546 + 0.139i)17-s + (−0.5 + 0.866i)18-s + (−0.826 − 1.43i)19-s + (1.19 + 1.49i)22-s + (0.0747 − 0.997i)25-s + (0.365 − 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.977 - 0.212i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.977 - 0.212i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010006030\)
\(L(\frac12)\) \(\approx\) \(1.010006030\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0747 - 0.997i)T \)
7 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good3 \( 1 + (-0.826 - 0.563i)T^{2} \)
5 \( 1 + (-0.0747 + 0.997i)T^{2} \)
11 \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \)
17 \( 1 + (-0.0546 - 0.139i)T + (-0.733 + 0.680i)T^{2} \)
19 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.109 + 1.46i)T + (-0.988 + 0.149i)T^{2} \)
53 \( 1 + (-0.988 + 0.149i)T + (0.955 - 0.294i)T^{2} \)
59 \( 1 + (-1.44 - 1.34i)T + (0.0747 + 0.997i)T^{2} \)
61 \( 1 + (-0.988 - 0.149i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.878621745802636728788891431286, −8.461097435977670659429394275630, −7.24901257095510785850572857909, −6.90348708548079882240928186767, −6.18413420867010203574781878231, −5.36161597247590528138898838156, −4.29758341834531492106481643156, −3.80549888598504944190909918930, −2.57874050872139928251707585252, −0.73127102821152655947495634610, 1.35785563287857960668284014653, 2.21748602592232467163744945195, 3.53458468761138345313533954961, 4.01702667160260805116428061367, 4.79289818403610463006266760161, 6.01542984994861478371212139108, 6.72762075619863230819554512864, 7.44738125773912738451990304757, 8.669071028518762798510947781334, 9.392031329014663710697504276124

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