Properties

Label 2-6034-1.1-c1-0-112
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 − 2.30·3-s + 4-s − 2.85·5-s − 2.30·6-s + 7-s + 8-s + 2.31·9-s − 2.85·10-s − 1.11·11-s − 2.30·12-s + 2.44·13-s + 14-s + 6.57·15-s + 16-s + 1.74·17-s + 2.31·18-s − 0.0622·19-s − 2.85·20-s − 2.30·21-s − 1.11·22-s − 4.67·23-s − 2.30·24-s + 3.13·25-s + 2.44·26-s + 1.57·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.33·3-s + 0.5·4-s − 1.27·5-s − 0.941·6-s + 0.377·7-s + 0.353·8-s + 0.772·9-s − 0.902·10-s − 0.336·11-s − 0.665·12-s + 0.679·13-s + 0.267·14-s + 1.69·15-s + 0.250·16-s + 0.422·17-s + 0.546·18-s − 0.0142·19-s − 0.637·20-s − 0.503·21-s − 0.238·22-s − 0.974·23-s − 0.470·24-s + 0.627·25-s + 0.480·26-s + 0.302·27-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 \( 1 + 2.30T + 3T^{2} \)
5 \( 1 + 2.85T + 5T^{2} \)
11 \( 1 + 1.11T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 - 1.74T + 17T^{2} \)
19 \( 1 + 0.0622T + 19T^{2} \)
23 \( 1 + 4.67T + 23T^{2} \)
29 \( 1 + 9.83T + 29T^{2} \)
31 \( 1 + 0.366T + 31T^{2} \)
37 \( 1 + 1.57T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 1.33T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 3.40T + 53T^{2} \)
59 \( 1 - 6.27T + 59T^{2} \)
61 \( 1 - 2.39T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 - 6.35T + 71T^{2} \)
73 \( 1 - 3.56T + 73T^{2} \)
79 \( 1 - 8.90T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 6.43T + 89T^{2} \)
97 \( 1 + 0.643T + 97T^{2} \)
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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

−7.54414860052682142315478164130, −7.01412266847184265202089711941, −5.88780640204769767780795182378, −5.74974462806835990837227971185, −4.84665560701640408378332423208, −4.05865769024692710972488094700, −3.67145333087341656666346817176, −2.40326998126095232235555006964, −1.10923141889783700172648808879, 0, 1.10923141889783700172648808879, 2.40326998126095232235555006964, 3.67145333087341656666346817176, 4.05865769024692710972488094700, 4.84665560701640408378332423208, 5.74974462806835990837227971185, 5.88780640204769767780795182378, 7.01412266847184265202089711941, 7.54414860052682142315478164130

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