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Properties

Label 2-1548-129.128-c1-0-9
Degree $2$
Conductor $1548$
Sign $0.577 + 0.816i$
Analytic cond. $12.3608$
Root an. cond. $3.51579$
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  + 0.731i·11-s + 0.680·13-s − 7.62i·17-s − 4.52i·23-s − 5·25-s + 11.0·31-s + 12.3i·41-s + 6.55·43-s − 6.44i·47-s + 7·49-s − 10.8i·53-s − 14.9i·59-s − 9.71·67-s + 13.1·79-s − 10.7i·83-s + ⋯
L(s)  = 1  + 0.220i·11-s + 0.188·13-s − 1.85i·17-s − 0.942i·23-s − 25-s + 1.98·31-s + 1.92i·41-s + 1.00·43-s − 0.940i·47-s + 49-s − 1.49i·53-s − 1.94i·59-s − 1.18·67-s + 1.47·79-s − 1.17i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1548\)    =    \(2^{2} \cdot 3^{2} \cdot 43\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(12.3608\)
Root analytic conductor: \(3.51579\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1548} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1548,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.545061117\)
\(L(\frac12)\) \(\approx\) \(1.545061117\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
43 \( 1 - 6.55T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 0.731iT - 11T^{2} \)
13 \( 1 - 0.680T + 13T^{2} \)
17 \( 1 + 7.62iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 4.52iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 11.0T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
47 \( 1 + 6.44iT - 47T^{2} \)
53 \( 1 + 10.8iT - 53T^{2} \)
59 \( 1 + 14.9iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 9.71T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 10.7iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−9.481111187615601600870004211058, −8.469367289983601473789260047183, −7.77833280554880709708366138154, −6.86988487461013685244510943425, −6.16942050425981871929221199552, −5.05305724547585691893125764724, −4.40809102685704376428481502467, −3.16853335348086635905557679747, −2.25164979758916796444015130044, −0.67505974055670308882709633948, 1.24903183905978761623822440843, 2.47802694259507811610323396838, 3.71230916128562033798352951313, 4.36874529898802824097721433309, 5.74317201938704497384553104875, 6.08172049500662781696750373234, 7.25855776819221501959087029684, 8.006523172555836040757843936686, 8.732880053364530602812810251640, 9.532223316812814844605419843625

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