Properties

Label 2-3042-13.12-c1-0-53
Degree $2$
Conductor $3042$
Sign $-0.960 + 0.277i$
Analytic cond. $24.2904$
Root an. cond. $4.92853$
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  i·2-s − 4-s − 1.73i·5-s + 4.73i·7-s + i·8-s − 1.73·10-s − 4.73i·11-s + 4.73·14-s + 16-s − 5.19·17-s − 1.26i·19-s + 1.73i·20-s − 4.73·22-s + 2.19·23-s + 2.00·25-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.774i·5-s + 1.78i·7-s + 0.353i·8-s − 0.547·10-s − 1.42i·11-s + 1.26·14-s + 0.250·16-s − 1.26·17-s − 0.290i·19-s + 0.387i·20-s − 1.00·22-s + 0.457·23-s + 0.400·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3042\)    =    \(2 \cdot 3^{2} \cdot 13^{2}\)
Sign: $-0.960 + 0.277i$
Analytic conductor: \(24.2904\)
Root analytic conductor: \(4.92853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3042} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3042,\ (\ :1/2),\ -0.960 + 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9261606618\)
\(L(\frac12)\) \(\approx\) \(0.9261606618\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 - 4.73iT - 7T^{2} \)
11 \( 1 + 4.73iT - 11T^{2} \)
17 \( 1 + 5.19T + 17T^{2} \)
19 \( 1 + 1.26iT - 19T^{2} \)
23 \( 1 - 2.19T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 2.53iT - 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 + 0.464iT - 41T^{2} \)
43 \( 1 + 6.19T + 43T^{2} \)
47 \( 1 - 1.26iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + 13.8iT - 59T^{2} \)
61 \( 1 - 4.80T + 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 - 8.19iT - 71T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + 2.53iT - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−8.580080998641710208433201245408, −8.151216046749202505879480909086, −6.63320076716790490822361960942, −5.94968681617951073683584634279, −5.15967365969020026069904649934, −4.60839319173876106184715581880, −3.31088023549051013013276678955, −2.64823604294301214148879193302, −1.66520142302483777679319300576, −0.30300188091268477826440440497, 1.26387557611753727194401765180, 2.61812365475114739270998892494, 3.82078439526043869015774741772, 4.38386767823403846317582874454, 5.10717238815327686419250091804, 6.43710574658513813850490025565, 7.00841506986143513427241358296, 7.18958337044036658506612967423, 8.081654031349349833089996492157, 8.990815706107538061250050632790

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