Properties

Label 2-6034-1.1-c1-0-98
Degree $2$
Conductor $6034$
Sign $1$
Analytic cond. $48.1817$
Root an. cond. $6.94130$
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 + 0.630·3-s + 4-s + 3.47·5-s + 0.630·6-s − 7-s + 8-s − 2.60·9-s + 3.47·10-s − 2.00·11-s + 0.630·12-s − 1.94·13-s − 14-s + 2.19·15-s + 16-s + 3.57·17-s − 2.60·18-s + 7.18·19-s + 3.47·20-s − 0.630·21-s − 2.00·22-s − 3.84·23-s + 0.630·24-s + 7.07·25-s − 1.94·26-s − 3.53·27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.364·3-s + 0.5·4-s + 1.55·5-s + 0.257·6-s − 0.377·7-s + 0.353·8-s − 0.867·9-s + 1.09·10-s − 0.603·11-s + 0.182·12-s − 0.539·13-s − 0.267·14-s + 0.565·15-s + 0.250·16-s + 0.865·17-s − 0.613·18-s + 1.64·19-s + 0.777·20-s − 0.137·21-s − 0.426·22-s − 0.801·23-s + 0.128·24-s + 1.41·25-s − 0.381·26-s − 0.679·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6034\)    =    \(2 \cdot 7 \cdot 431\)
Sign: $1$
Analytic conductor: \(48.1817\)
Root analytic conductor: \(6.94130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.602740257\)
\(L(\frac12)\) \(\approx\) \(4.602740257\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + T \)
431 \( 1 + T \)
good3 \( 1 - 0.630T + 3T^{2} \)
5 \( 1 - 3.47T + 5T^{2} \)
11 \( 1 + 2.00T + 11T^{2} \)
13 \( 1 + 1.94T + 13T^{2} \)
17 \( 1 - 3.57T + 17T^{2} \)
19 \( 1 - 7.18T + 19T^{2} \)
23 \( 1 + 3.84T + 23T^{2} \)
29 \( 1 - 7.36T + 29T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 - 8.12T + 37T^{2} \)
41 \( 1 - 6.82T + 41T^{2} \)
43 \( 1 - 5.87T + 43T^{2} \)
47 \( 1 - 6.49T + 47T^{2} \)
53 \( 1 + 4.20T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + 5.84T + 61T^{2} \)
67 \( 1 - 2.15T + 67T^{2} \)
71 \( 1 + 0.416T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + 1.88T + 89T^{2} \)
97 \( 1 + 7.27T + 97T^{2} \)
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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

−7.83671019652302985507352794195, −7.45655293729456986421791282508, −6.29264254971916714114853996649, −5.86550496697101641937520596936, −5.38436192690207679492764705782, −4.61451806188428044539812000356, −3.38239588930568265942520052387, −2.75122661452129597597845874730, −2.24949302748835041123076214188, −1.01713697055146651439918183417, 1.01713697055146651439918183417, 2.24949302748835041123076214188, 2.75122661452129597597845874730, 3.38239588930568265942520052387, 4.61451806188428044539812000356, 5.38436192690207679492764705782, 5.86550496697101641937520596936, 6.29264254971916714114853996649, 7.45655293729456986421791282508, 7.83671019652302985507352794195

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