Properties

Label 2-2548-2548.155-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  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)8-s + (0.623 − 0.781i)9-s + (1.12 + 1.40i)11-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s + (−0.277 − 1.21i)17-s − 0.999·18-s − 1.24·19-s + (0.400 − 1.75i)22-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)8-s + (0.623 − 0.781i)9-s + (1.12 + 1.40i)11-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s + (−0.277 − 1.21i)17-s − 0.999·18-s − 1.24·19-s + (0.400 − 1.75i)22-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)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: $\chi_{2548} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9726674775\)
\(L(\frac12)\) \(\approx\) \(0.9726674775\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.623 - 0.781i)T \)
good3 \( 1 + (-0.623 + 0.781i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
19 \( 1 + 1.24T + T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
29 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + 1.24T + T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.445 - 1.94i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \)
61 \( 1 + (0.445 + 1.94i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 - 0.445T + T^{2} \)
71 \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.222 + 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
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

−9.056524198829664815906161117751, −8.907936083124650340922509049796, −7.61905587609909426986840796107, −6.84044724117511008089653924367, −6.35674388255373662121523280369, −4.73445706412136860626948757396, −4.33252680528557991642964726779, −3.27091258126448816161472248267, −2.13710632925934534562561376081, −1.41239365191081272110012516078, 0.912869399642643746034070220470, 1.92235456047270173085048507721, 3.74450063105472073156343358942, 4.23208534018140589103589800234, 5.41414831405643461223343703261, 6.15603124082345375193345577441, 6.79329934569825745118713322216, 7.61913556534681115613232340068, 8.426168979198882412168802301938, 8.664310511281954165098725356483

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