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Properties

Label 2-2548-13.3-c1-0-7
Degree $2$
Conductor $2548$
Sign $0.0128 - 0.999i$
Analytic cond. $20.3458$
Root an. cond. $4.51064$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + (1.5 − 2.59i)9-s + (1 + 1.73i)11-s + (−3.5 + 0.866i)13-s + (−1.5 + 2.59i)17-s + (−3 + 5.19i)19-s + (2 + 3.46i)23-s − 4·25-s + (3.5 + 6.06i)29-s − 4·31-s + (−4.5 − 7.79i)37-s + (4.5 + 7.79i)41-s + (−5 + 8.66i)43-s + (1.5 − 2.59i)45-s + 2·47-s + ⋯
L(s)  = 1  + 0.447·5-s + (0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s + (−0.970 + 0.240i)13-s + (−0.363 + 0.630i)17-s + (−0.688 + 1.19i)19-s + (0.417 + 0.722i)23-s − 0.800·25-s + (0.649 + 1.12i)29-s − 0.718·31-s + (−0.739 − 1.28i)37-s + (0.702 + 1.21i)41-s + (−0.762 + 1.32i)43-s + (0.223 − 0.387i)45-s + 0.291·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.428628038\)
\(L(\frac12)\) \(\approx\) \(1.428628038\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 + (3.5 - 0.866i)T \)
good3 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (4.5 + 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5 - 8.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.237345327956482721394767972710, −8.392251054166184215278769066262, −7.43196520287299727545806214717, −6.79723032020176407291172815408, −6.07640451188673910534696769876, −5.20859730866804991322825135776, −4.21834833239435695751417974752, −3.55765056546726034712774659494, −2.22336974564856587144707062007, −1.38768714962326992062310427925, 0.46032145690407628496447076092, 2.07222224257763937184449827180, 2.64698627841450364886970071717, 3.99833159966872107770044250378, 4.85239784626783516568776957328, 5.44206600500674848502534607058, 6.54261187815172419624591523247, 7.11196513489318938581663957820, 7.931956435518194900755251474421, 8.797434169369425914973336238245

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