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Properties

Label 2-2548-364.107-c0-0-3
Degree $2$
Conductor $2548$
Sign $0.617 + 0.786i$
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  + 2-s + 4-s + (0.5 − 0.866i)5-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)13-s + 16-s − 17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)26-s + (0.5 + 0.866i)29-s + 32-s − 34-s + ⋯
L(s)  = 1  + 2-s + 4-s + (0.5 − 0.866i)5-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)13-s + 16-s − 17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)26-s + (0.5 + 0.866i)29-s + 32-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.617 + 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.367232227\)
\(L(\frac12)\) \(\approx\) \(2.367232227\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 \)
13 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.927985960370599666662731004970, −8.260647789872343148639306205530, −7.20966724218635275576711196067, −6.50102174360955948750965746306, −5.67291742724698056085365586391, −5.12175478759810455723173361593, −4.30880071551148722300510231442, −3.29596623432318397699729590580, −2.45337349565767992173666029371, −1.18539136044969625419897723988, 2.11010312278356418964214055078, 2.40179397787157851373986257058, 3.54014168293604845347469784919, 4.53957051526814813644086943945, 5.20600763774967838466757202606, 6.16141182908028041210208357865, 6.70388492245890299276303106858, 7.39016527174739922299213375840, 8.276886126852527558802927216605, 9.241412731669405311842006834279

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