Properties

Label 2-6034-1.1-c1-0-201
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.00·3-s + 4-s + 2.56·5-s − 2.00·6-s + 7-s − 8-s + 1.03·9-s − 2.56·10-s − 2.73·11-s + 2.00·12-s − 1.28·13-s − 14-s + 5.14·15-s + 16-s − 3.83·17-s − 1.03·18-s − 5.83·19-s + 2.56·20-s + 2.00·21-s + 2.73·22-s − 4.22·23-s − 2.00·24-s + 1.55·25-s + 1.28·26-s − 3.94·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.16·3-s + 0.5·4-s + 1.14·5-s − 0.820·6-s + 0.377·7-s − 0.353·8-s + 0.345·9-s − 0.809·10-s − 0.824·11-s + 0.580·12-s − 0.357·13-s − 0.267·14-s + 1.32·15-s + 0.250·16-s − 0.931·17-s − 0.244·18-s − 1.33·19-s + 0.572·20-s + 0.438·21-s + 0.582·22-s − 0.881·23-s − 0.410·24-s + 0.311·25-s + 0.252·26-s − 0.758·27-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: \(1\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.00T + 3T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
17 \( 1 + 3.83T + 17T^{2} \)
19 \( 1 + 5.83T + 19T^{2} \)
23 \( 1 + 4.22T + 23T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
31 \( 1 - 0.0674T + 31T^{2} \)
37 \( 1 - 9.14T + 37T^{2} \)
41 \( 1 - 3.90T + 41T^{2} \)
43 \( 1 - 3.92T + 43T^{2} \)
47 \( 1 + 7.32T + 47T^{2} \)
53 \( 1 + 1.13T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 0.938T + 71T^{2} \)
73 \( 1 + 5.30T + 73T^{2} \)
79 \( 1 + 5.57T + 79T^{2} \)
83 \( 1 - 4.27T + 83T^{2} \)
89 \( 1 + 1.86T + 89T^{2} \)
97 \( 1 + 0.460T + 97T^{2} \)
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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

−7.82409300044693863622691773746, −7.37860052483998077314295292213, −6.22173215402598162435924269146, −5.88082823093622105331740451901, −4.78997192147272741035524533053, −3.92987138315771062566836085053, −2.74671614012987942046776675135, −2.24274013617508248506204718051, −1.73904111707420843247577497946, 0, 1.73904111707420843247577497946, 2.24274013617508248506204718051, 2.74671614012987942046776675135, 3.92987138315771062566836085053, 4.78997192147272741035524533053, 5.88082823093622105331740451901, 6.22173215402598162435924269146, 7.37860052483998077314295292213, 7.82409300044693863622691773746

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