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Properties

Label 2-2548-2548.519-c0-0-0
Degree $2$
Conductor $2548$
Sign $-0.672 - 0.740i$
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  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (0.400 + 0.193i)11-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + (−1.12 + 1.40i)17-s + 18-s − 1.80·19-s + (−0.277 − 0.347i)22-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (0.400 + 0.193i)11-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + (−1.12 + 1.40i)17-s + 18-s − 1.80·19-s + (−0.277 − 0.347i)22-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.672 - 0.740i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ -0.672 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1409559136\)
\(L(\frac12)\) \(\approx\) \(0.1409559136\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good3 \( 1 + (0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
29 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + 1.80T + T^{2} \)
37 \( 1 + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (-1.24 - 1.56i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
61 \( 1 + (-1.24 + 1.56i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 - 1.24T + T^{2} \)
71 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.623 + 0.781i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
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

−9.204095416924406886021553864739, −8.646640444607979908814621735041, −7.973914685989337889762533050598, −7.32454006610847490836459373662, −6.51014440698232011637418489968, −5.57514429897658368553406083173, −4.33270689296825664965802045769, −3.74777324965756976838710035625, −2.35740084933941819013462780024, −1.75687073306799504209583315721, 0.11046105071271163764349690356, 2.08459584971749161791390679491, 2.48049485506988077784329444238, 4.10538103877434369000200204310, 5.15728667979485409649516421669, 5.79713883030326099742629988677, 6.66071455990545131474424310348, 7.24997485836878967716831416475, 8.320256920633590350319528072447, 8.775259642494164314490370382506

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