Properties

Label 2-6034-1.1-c1-0-175
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 + 1.20·3-s + 4-s + 2.17·5-s − 1.20·6-s + 7-s − 8-s − 1.54·9-s − 2.17·10-s − 1.32·11-s + 1.20·12-s − 6.95·13-s − 14-s + 2.61·15-s + 16-s + 5.76·17-s + 1.54·18-s − 3.84·19-s + 2.17·20-s + 1.20·21-s + 1.32·22-s − 2.41·23-s − 1.20·24-s − 0.286·25-s + 6.95·26-s − 5.48·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.696·3-s + 0.5·4-s + 0.970·5-s − 0.492·6-s + 0.377·7-s − 0.353·8-s − 0.515·9-s − 0.686·10-s − 0.399·11-s + 0.348·12-s − 1.92·13-s − 0.267·14-s + 0.676·15-s + 0.250·16-s + 1.39·17-s + 0.364·18-s − 0.882·19-s + 0.485·20-s + 0.263·21-s + 0.282·22-s − 0.502·23-s − 0.246·24-s − 0.0572·25-s + 1.36·26-s − 1.05·27-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 \( 1 - 1.20T + 3T^{2} \)
5 \( 1 - 2.17T + 5T^{2} \)
11 \( 1 + 1.32T + 11T^{2} \)
13 \( 1 + 6.95T + 13T^{2} \)
17 \( 1 - 5.76T + 17T^{2} \)
19 \( 1 + 3.84T + 19T^{2} \)
23 \( 1 + 2.41T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + 9.33T + 31T^{2} \)
37 \( 1 - 5.03T + 37T^{2} \)
41 \( 1 - 2.43T + 41T^{2} \)
43 \( 1 + 3.38T + 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 - 5.31T + 53T^{2} \)
59 \( 1 - 6.17T + 59T^{2} \)
61 \( 1 + 7.19T + 61T^{2} \)
67 \( 1 + 2.92T + 67T^{2} \)
71 \( 1 + 8.33T + 71T^{2} \)
73 \( 1 - 5.89T + 73T^{2} \)
79 \( 1 + 6.68T + 79T^{2} \)
83 \( 1 - 5.71T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 9.37T + 97T^{2} \)
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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

−7.85889928064333122079333160842, −7.30338642746301742575682556402, −6.36111303698857877567930435627, −5.57315376130094779176786464476, −5.06185865616799025400991328317, −3.90976240878098516121683535139, −2.61408191077566149404711631456, −2.52376520470006870206893663622, −1.48126606591353479391990498375, 0, 1.48126606591353479391990498375, 2.52376520470006870206893663622, 2.61408191077566149404711631456, 3.90976240878098516121683535139, 5.06185865616799025400991328317, 5.57315376130094779176786464476, 6.36111303698857877567930435627, 7.30338642746301742575682556402, 7.85889928064333122079333160842

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