Properties

Label 2-2548-2548.1299-c0-0-0
Degree $2$
Conductor $2548$
Sign $0.942 + 0.335i$
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  + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.826 − 0.563i)7-s + (−0.623 + 0.781i)8-s + (−0.733 + 0.680i)9-s + (0.535 + 0.496i)11-s + (−0.222 + 0.974i)13-s + (0.955 + 0.294i)14-s + (0.365 − 0.930i)16-s + (0.142 − 1.90i)17-s + (0.5 − 0.866i)18-s + (−0.733 − 1.26i)19-s + (−0.658 − 0.317i)22-s + (0.955 + 0.294i)25-s + (−0.0747 − 0.997i)26-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.826 − 0.563i)7-s + (−0.623 + 0.781i)8-s + (−0.733 + 0.680i)9-s + (0.535 + 0.496i)11-s + (−0.222 + 0.974i)13-s + (0.955 + 0.294i)14-s + (0.365 − 0.930i)16-s + (0.142 − 1.90i)17-s + (0.5 − 0.866i)18-s + (−0.733 − 1.26i)19-s + (−0.658 − 0.317i)22-s + (0.955 + 0.294i)25-s + (−0.0747 − 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.942 + 0.335i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.942 + 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6398259093\)
\(L(\frac12)\) \(\approx\) \(0.6398259093\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.955 - 0.294i)T \)
7 \( 1 + (0.826 + 0.563i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
good3 \( 1 + (0.733 - 0.680i)T^{2} \)
5 \( 1 + (-0.955 - 0.294i)T^{2} \)
11 \( 1 + (-0.535 - 0.496i)T + (0.0747 + 0.997i)T^{2} \)
17 \( 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2} \)
19 \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.988 - 0.149i)T^{2} \)
29 \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-1.88 + 0.582i)T + (0.826 - 0.563i)T^{2} \)
53 \( 1 + (0.826 - 0.563i)T + (0.365 - 0.930i)T^{2} \)
59 \( 1 + (-1.63 + 0.246i)T + (0.955 - 0.294i)T^{2} \)
61 \( 1 + (0.826 + 0.563i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.72 - 0.829i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.0747 + 0.997i)T^{2} \)
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

−9.130719061672923043685370450122, −8.436652379702907805273509535420, −7.32335894474094311180186224056, −6.93695773489676212480276789512, −6.35069250027195575477413305445, −5.15336896968513287155707911905, −4.46079976208898337606423269269, −2.93300900298812348688110475679, −2.32760106054431835396629145538, −0.71031078871356481121433447488, 1.02954673507593643672465603530, 2.43129427400672876254454075351, 3.28721983630939741140046362539, 3.90030967976046433439745280913, 5.63798784780453617774746441420, 6.21961740792683510724527580187, 6.68834055631876256465597611621, 7.985121330669656893288118695583, 8.540829906690201730051119147615, 8.919125958457076623972746596701

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