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Properties

Label 2-2548-2548.1299-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(2.304433979\)
\(L(\frac12)\) \(\approx\) \(2.304433979\)
\(L(1)\) 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

−9.120501312962982343081846914995, −8.048335851856443403772199862536, −7.60951388305992355916810560954, −6.49609843589026195974965071102, −5.74559032326171107026251265849, −4.90428173631263898528872768577, −4.64753335472094200359146387352, −3.07204438003778353401975386756, −2.63772142143068538801657371568, −1.46400368331919978083130565603, 1.42390388948599097099699868207, 2.83277033916284024207966947790, 3.38704957839301527204835072913, 4.67816652095372992283696033666, 4.97028628686637405404636467804, 6.01348298970156180172280594715, 6.70046322580310547249518656732, 7.53003405052503288737192255246, 8.254227424394810061638953926863, 8.750823325449607362870152121600

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