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Properties

Label 2-2548-2548.1611-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: $\chi_{2548} (1611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.545084753\)
\(L(\frac12)\) \(\approx\) \(1.545084753\)
\(L(1)\) 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

−9.347493299488856606994412847748, −8.260026869187664676291449617500, −7.68171256857423161494723088455, −6.53303167761283146604147286322, −5.72985503460696697897617583354, −5.01380273599690497579996983329, −4.41497608764527238066639791605, −3.18842402742275149932823333015, −2.43076141179881623217973194133, −1.54903126960198049145235450077, 0.897291439800504648177859578925, 2.99445162852322899791826855110, 3.39936691225553138115931861182, 4.48688613685936144213388161352, 5.11878894831646751383773101079, 6.15161730197122545312590826588, 6.80253659716366134984994813288, 7.32916772315651483885888172434, 8.233871203977983600480714057244, 8.897611847811135928169351972244

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