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Properties

Label 2-2548-2548.935-c0-0-0
Degree $2$
Conductor $2548$
Sign $-0.304 - 0.952i$
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  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.365 + 0.930i)7-s + (0.222 − 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.109 − 1.46i)11-s + (−0.900 + 0.433i)13-s + (0.826 − 0.563i)14-s + (−0.733 + 0.680i)16-s + (−1.63 + 0.246i)17-s + (0.5 − 0.866i)18-s + (0.0747 + 0.129i)19-s + (−0.914 + 1.14i)22-s + (0.826 − 0.563i)25-s + (0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.365 + 0.930i)7-s + (0.222 − 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.109 − 1.46i)11-s + (−0.900 + 0.433i)13-s + (0.826 − 0.563i)14-s + (−0.733 + 0.680i)16-s + (−1.63 + 0.246i)17-s + (0.5 − 0.866i)18-s + (0.0747 + 0.129i)19-s + (−0.914 + 1.14i)22-s + (0.826 − 0.563i)25-s + (0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.304 - 0.952i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (935, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ -0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3989276386\)
\(L(\frac12)\) \(\approx\) \(0.3989276386\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
good3 \( 1 + (-0.0747 - 0.997i)T^{2} \)
5 \( 1 + (-0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.109 + 1.46i)T + (-0.988 - 0.149i)T^{2} \)
17 \( 1 + (1.63 - 0.246i)T + (0.955 - 0.294i)T^{2} \)
19 \( 1 + (-0.0747 - 0.129i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.955 - 0.294i)T^{2} \)
29 \( 1 + (-1.19 - 1.49i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (1.57 + 1.07i)T + (0.365 + 0.930i)T^{2} \)
53 \( 1 + (0.365 + 0.930i)T + (-0.733 + 0.680i)T^{2} \)
59 \( 1 + (0.698 + 0.215i)T + (0.826 + 0.563i)T^{2} \)
61 \( 1 + (0.365 - 0.930i)T + (-0.733 - 0.680i)T^{2} \)
67 \( 1 + (0.988 - 1.71i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (1.03 - 1.29i)T + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
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

−8.992744637470260723363354276647, −8.751202292713443950762873515473, −8.184216178863053965903348778052, −6.95786576487147179803209341790, −6.58043438225980940952312266605, −5.35119639367540302841538910889, −4.55138656650202942187605861975, −3.24351441504279358667770621491, −2.61970004669131806992988833852, −1.64212138597743798653548030918, 0.32732673443606019708119674068, 1.80208442575299469267476657626, 2.91324341145270040845542747667, 4.40301438212980905864951612452, 4.75486255014698663597513241731, 6.24862773781326545348192040341, 6.59955771430853612321336837292, 7.44759394465055014578215380637, 7.80649974890830825813950603373, 9.208473984812952177201293208253

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