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Properties

Label 2-2548-2548.2391-c0-0-1
Degree $2$
Conductor $2548$
Sign $0.991 + 0.127i$
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  + (0.988 + 0.149i)2-s + (0.955 + 0.294i)4-s + (−0.955 + 0.294i)7-s + (0.900 + 0.433i)8-s + (0.365 − 0.930i)9-s + (−0.603 − 1.53i)11-s + (0.623 − 0.781i)13-s + (−0.988 + 0.149i)14-s + (0.826 + 0.563i)16-s + (1.44 + 1.34i)17-s + (0.5 − 0.866i)18-s + (0.365 + 0.632i)19-s + (−0.367 − 1.61i)22-s + (−0.988 + 0.149i)25-s + (0.733 − 0.680i)26-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)2-s + (0.955 + 0.294i)4-s + (−0.955 + 0.294i)7-s + (0.900 + 0.433i)8-s + (0.365 − 0.930i)9-s + (−0.603 − 1.53i)11-s + (0.623 − 0.781i)13-s + (−0.988 + 0.149i)14-s + (0.826 + 0.563i)16-s + (1.44 + 1.34i)17-s + (0.5 − 0.866i)18-s + (0.365 + 0.632i)19-s + (−0.367 − 1.61i)22-s + (−0.988 + 0.149i)25-s + (0.733 − 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.991 + 0.127i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (2391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.212097904\)
\(L(\frac12)\) \(\approx\) \(2.212097904\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good3 \( 1 + (-0.365 + 0.930i)T^{2} \)
5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (0.603 + 1.53i)T + (-0.733 + 0.680i)T^{2} \)
17 \( 1 + (-1.44 - 1.34i)T + (0.0747 + 0.997i)T^{2} \)
19 \( 1 + (-0.365 - 0.632i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.826 + 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.147 - 0.0222i)T + (0.955 + 0.294i)T^{2} \)
53 \( 1 + (0.955 + 0.294i)T + (0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.142 - 1.90i)T + (-0.988 - 0.149i)T^{2} \)
61 \( 1 + (0.955 - 0.294i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.733 - 1.26i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.440 + 1.92i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
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

−8.972967447936376180987917746023, −8.058016434596323496550070673416, −7.60008913644109625247718096556, −6.23022421457611098764913784903, −6.00835690616654244171502937415, −5.52034112792694511585203062802, −3.96730298015779383820196415333, −3.46196044576433168294581763028, −2.88782054821610161439497488278, −1.23756685469060614001044102641, 1.56081908799475983762261108125, 2.59964822876328798820147297673, 3.40973535914702531617421748798, 4.50788917704683282108143243391, 4.95758960295658123366529046252, 5.88790626095213418103277857495, 6.90309546054638711628279410271, 7.28533106152279362867221186976, 7.992890790357884476479579062709, 9.566821058687864558430909115179

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