Properties

Label 2-6045-1.1-c1-0-184
Degree $2$
Conductor $6045$
Sign $-1$
Analytic cond. $48.2695$
Root an. cond. $6.94763$
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  − 0.139·2-s + 3-s − 1.98·4-s + 5-s − 0.139·6-s − 0.572·7-s + 0.553·8-s + 9-s − 0.139·10-s − 2.21·11-s − 1.98·12-s − 13-s + 0.0796·14-s + 15-s + 3.88·16-s + 4.50·17-s − 0.139·18-s − 3.16·19-s − 1.98·20-s − 0.572·21-s + 0.308·22-s − 5.35·23-s + 0.553·24-s + 25-s + 0.139·26-s + 27-s + 1.13·28-s + ⋯
L(s)  = 1  − 0.0983·2-s + 0.577·3-s − 0.990·4-s + 0.447·5-s − 0.0567·6-s − 0.216·7-s + 0.195·8-s + 0.333·9-s − 0.0439·10-s − 0.668·11-s − 0.571·12-s − 0.277·13-s + 0.0212·14-s + 0.258·15-s + 0.971·16-s + 1.09·17-s − 0.0327·18-s − 0.725·19-s − 0.442·20-s − 0.125·21-s + 0.0657·22-s − 1.11·23-s + 0.112·24-s + 0.200·25-s + 0.0272·26-s + 0.192·27-s + 0.214·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
Sign: $-1$
Analytic conductor: \(48.2695\)
Root analytic conductor: \(6.94763\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6045,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 + T \)
31 \( 1 + T \)
good2 \( 1 + 0.139T + 2T^{2} \)
7 \( 1 + 0.572T + 7T^{2} \)
11 \( 1 + 2.21T + 11T^{2} \)
17 \( 1 - 4.50T + 17T^{2} \)
19 \( 1 + 3.16T + 19T^{2} \)
23 \( 1 + 5.35T + 23T^{2} \)
29 \( 1 + 4.19T + 29T^{2} \)
37 \( 1 - 8.21T + 37T^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 - 8.80T + 43T^{2} \)
47 \( 1 - 3.29T + 47T^{2} \)
53 \( 1 - 4.15T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 + 7.14T + 61T^{2} \)
67 \( 1 + 6.43T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 - 3.79T + 79T^{2} \)
83 \( 1 + 8.15T + 83T^{2} \)
89 \( 1 - 17.6T + 89T^{2} \)
97 \( 1 - 1.30T + 97T^{2} \)
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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

−7.77704864835371236834609107652, −7.36324043231271178792440493072, −5.96862106383675862593273223685, −5.76367426237502874629117196089, −4.65598389975385980260245095607, −4.12631534913538762255309510229, −3.20045335120152770057655355709, −2.40017831454501391042699709329, −1.31179856688213196032033022459, 0, 1.31179856688213196032033022459, 2.40017831454501391042699709329, 3.20045335120152770057655355709, 4.12631534913538762255309510229, 4.65598389975385980260245095607, 5.76367426237502874629117196089, 5.96862106383675862593273223685, 7.36324043231271178792440493072, 7.77704864835371236834609107652

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