Properties

Label 2-6026-1.1-c1-0-22
Degree $2$
Conductor $6026$
Sign $1$
Analytic cond. $48.1178$
Root an. cond. $6.93670$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.36·3-s + 4-s + 2.69·5-s − 3.36·6-s − 4.35·7-s + 8-s + 8.30·9-s + 2.69·10-s − 4.44·11-s − 3.36·12-s − 2.85·13-s − 4.35·14-s − 9.05·15-s + 16-s − 4.20·17-s + 8.30·18-s − 5.66·19-s + 2.69·20-s + 14.6·21-s − 4.44·22-s − 23-s − 3.36·24-s + 2.25·25-s − 2.85·26-s − 17.8·27-s − 4.35·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.94·3-s + 0.5·4-s + 1.20·5-s − 1.37·6-s − 1.64·7-s + 0.353·8-s + 2.76·9-s + 0.851·10-s − 1.33·11-s − 0.970·12-s − 0.790·13-s − 1.16·14-s − 2.33·15-s + 0.250·16-s − 1.01·17-s + 1.95·18-s − 1.29·19-s + 0.602·20-s + 3.19·21-s − 0.947·22-s − 0.208·23-s − 0.686·24-s + 0.450·25-s − 0.559·26-s − 3.43·27-s − 0.822·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6026\)    =    \(2 \cdot 23 \cdot 131\)
Sign: $1$
Analytic conductor: \(48.1178\)
Root analytic conductor: \(6.93670\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6026,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6723554156\)
\(L(\frac12)\) \(\approx\) \(0.6723554156\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
23 \( 1 + T \)
131 \( 1 + T \)
good3 \( 1 + 3.36T + 3T^{2} \)
5 \( 1 - 2.69T + 5T^{2} \)
7 \( 1 + 4.35T + 7T^{2} \)
11 \( 1 + 4.44T + 11T^{2} \)
13 \( 1 + 2.85T + 13T^{2} \)
17 \( 1 + 4.20T + 17T^{2} \)
19 \( 1 + 5.66T + 19T^{2} \)
29 \( 1 - 2.22T + 29T^{2} \)
31 \( 1 + 4.57T + 31T^{2} \)
37 \( 1 - 1.52T + 37T^{2} \)
41 \( 1 + 0.472T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + 2.55T + 47T^{2} \)
53 \( 1 + 6.65T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 + 8.36T + 61T^{2} \)
67 \( 1 - 4.53T + 67T^{2} \)
71 \( 1 + 7.21T + 71T^{2} \)
73 \( 1 + 9.18T + 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 - 6.45T + 83T^{2} \)
89 \( 1 + 4.86T + 89T^{2} \)
97 \( 1 - 2.32T + 97T^{2} \)
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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

−7.59881470748800318958520112531, −6.93125596271509567783951334731, −6.23171849275754103762738658226, −6.09418159264406157404947453746, −5.32973789168662850898111563385, −4.74460981087304251345020244034, −3.94153698763417416093873948313, −2.63314442291782372135866773681, −1.99574919658733635314274357459, −0.39726865027670081584152890086, 0.39726865027670081584152890086, 1.99574919658733635314274357459, 2.63314442291782372135866773681, 3.94153698763417416093873948313, 4.74460981087304251345020244034, 5.32973789168662850898111563385, 6.09418159264406157404947453746, 6.23171849275754103762738658226, 6.93125596271509567783951334731, 7.59881470748800318958520112531

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