Properties

Label 2-6026-1.1-c1-0-60
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 − 2.05·3-s + 4-s + 0.786·5-s − 2.05·6-s − 2.31·7-s + 8-s + 1.22·9-s + 0.786·10-s + 0.549·11-s − 2.05·12-s + 7.02·13-s − 2.31·14-s − 1.61·15-s + 16-s − 1.36·17-s + 1.22·18-s − 4.58·19-s + 0.786·20-s + 4.74·21-s + 0.549·22-s − 23-s − 2.05·24-s − 4.38·25-s + 7.02·26-s + 3.65·27-s − 2.31·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.18·3-s + 0.5·4-s + 0.351·5-s − 0.838·6-s − 0.873·7-s + 0.353·8-s + 0.407·9-s + 0.248·10-s + 0.165·11-s − 0.593·12-s + 1.94·13-s − 0.617·14-s − 0.417·15-s + 0.250·16-s − 0.330·17-s + 0.288·18-s − 1.05·19-s + 0.175·20-s + 1.03·21-s + 0.117·22-s − 0.208·23-s − 0.419·24-s − 0.876·25-s + 1.37·26-s + 0.702·27-s − 0.436·28-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

\(L(1)\) \(\approx\) \(1.835078665\)
\(L(\frac12)\) \(\approx\) \(1.835078665\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.05T + 3T^{2} \)
5 \( 1 - 0.786T + 5T^{2} \)
7 \( 1 + 2.31T + 7T^{2} \)
11 \( 1 - 0.549T + 11T^{2} \)
13 \( 1 - 7.02T + 13T^{2} \)
17 \( 1 + 1.36T + 17T^{2} \)
19 \( 1 + 4.58T + 19T^{2} \)
29 \( 1 + 4.02T + 29T^{2} \)
31 \( 1 - 0.353T + 31T^{2} \)
37 \( 1 - 5.31T + 37T^{2} \)
41 \( 1 - 3.51T + 41T^{2} \)
43 \( 1 - 2.95T + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 - 5.85T + 53T^{2} \)
59 \( 1 + 3.66T + 59T^{2} \)
61 \( 1 + 5.15T + 61T^{2} \)
67 \( 1 - 11.9T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + 1.84T + 73T^{2} \)
79 \( 1 + 4.22T + 79T^{2} \)
83 \( 1 + 1.45T + 83T^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 + 2.55T + 97T^{2} \)
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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

−7.965566069177766901174187412031, −6.95428416663205113313907332559, −6.29461343554168449923677247480, −5.95719659749760148203746218873, −5.55485333391160414412907411106, −4.32126899023355911317258977572, −3.92386703161388752167030150097, −2.91796178882406448561518133507, −1.84022158851888254800910233443, −0.67734363741562707448012324692, 0.67734363741562707448012324692, 1.84022158851888254800910233443, 2.91796178882406448561518133507, 3.92386703161388752167030150097, 4.32126899023355911317258977572, 5.55485333391160414412907411106, 5.95719659749760148203746218873, 6.29461343554168449923677247480, 6.95428416663205113313907332559, 7.965566069177766901174187412031

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