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Properties

Label 2-2548-2548.1143-c0-0-0
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  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.733 + 0.680i)7-s + (0.900 + 0.433i)8-s + (−0.988 + 0.149i)9-s + (0.147 + 0.0222i)11-s + (0.623 − 0.781i)13-s + (0.365 − 0.930i)14-s + (0.0747 − 0.997i)16-s + (0.698 − 0.215i)17-s + (0.5 + 0.866i)18-s + (−0.988 + 1.71i)19-s + (−0.0332 − 0.145i)22-s + (0.365 − 0.930i)25-s + (−0.955 − 0.294i)26-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.733 + 0.680i)7-s + (0.900 + 0.433i)8-s + (−0.988 + 0.149i)9-s + (0.147 + 0.0222i)11-s + (0.623 − 0.781i)13-s + (0.365 − 0.930i)14-s + (0.0747 − 0.997i)16-s + (0.698 − 0.215i)17-s + (0.5 + 0.866i)18-s + (−0.988 + 1.71i)19-s + (−0.0332 − 0.145i)22-s + (0.365 − 0.930i)25-s + (−0.955 − 0.294i)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: $\chi_{2548} (1143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9374678785\)
\(L(\frac12)\) \(\approx\) \(0.9374678785\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.365 + 0.930i)T \)
7 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good3 \( 1 + (0.988 - 0.149i)T^{2} \)
5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (-0.147 - 0.0222i)T + (0.955 + 0.294i)T^{2} \)
17 \( 1 + (-0.698 + 0.215i)T + (0.826 - 0.563i)T^{2} \)
19 \( 1 + (0.988 - 1.71i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.367 - 1.61i)T + (-0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.0747 - 0.997i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.603 + 1.53i)T + (-0.733 + 0.680i)T^{2} \)
53 \( 1 + (-0.733 + 0.680i)T + (0.0747 - 0.997i)T^{2} \)
59 \( 1 + (-1.21 - 0.825i)T + (0.365 + 0.930i)T^{2} \)
61 \( 1 + (-0.733 - 0.680i)T + (0.0747 + 0.997i)T^{2} \)
67 \( 1 + (-0.955 - 1.65i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.162 - 0.712i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.733 + 0.680i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.955 + 0.294i)T^{2} \)
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

−8.796016271156380090332521290091, −8.445578304427674660046240813036, −8.095639216094293473569284612376, −6.86951372051748784175846776454, −5.60314236522386151406093046130, −5.27731543773930862802630383416, −4.03275685629056108529636544139, −3.20909198258695443758784926168, −2.32362612448382412029237725297, −1.27435149612278647331941075899, 0.815061394441731962714200860872, 2.20150162273953135523329074705, 3.72292329885790707697653899952, 4.50144738516453024144190489990, 5.27316262437057300148016947518, 6.24287508561418797936763793561, 6.70248986421939493605076509647, 7.76209784534037901992605651020, 8.162809849775344893322618321823, 9.049343098147780894910521298385

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