Properties

Label 2-5850-1.1-c1-0-27
Degree $2$
Conductor $5850$
Sign $1$
Analytic cond. $46.7124$
Root an. cond. $6.83465$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 6·11-s + 13-s + 16-s + 6·19-s − 6·22-s − 6·23-s − 26-s − 2·29-s + 4·31-s − 32-s + 10·37-s − 6·38-s + 6·41-s + 8·43-s + 6·44-s + 6·46-s − 8·47-s − 7·49-s + 52-s + 6·53-s + 2·58-s − 10·59-s − 6·61-s − 4·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.80·11-s + 0.277·13-s + 1/4·16-s + 1.37·19-s − 1.27·22-s − 1.25·23-s − 0.196·26-s − 0.371·29-s + 0.718·31-s − 0.176·32-s + 1.64·37-s − 0.973·38-s + 0.937·41-s + 1.21·43-s + 0.904·44-s + 0.884·46-s − 1.16·47-s − 49-s + 0.138·52-s + 0.824·53-s + 0.262·58-s − 1.30·59-s − 0.768·61-s − 0.508·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5850\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(46.7124\)
Root analytic conductor: \(6.83465\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.745457698\)
\(L(\frac12)\) \(\approx\) \(1.745457698\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.914860535659067241209585074300, −7.70302577369861880861944948021, −6.49453266589921559642006169883, −6.34812805336864913009556585870, −5.37825277026734998627498746353, −4.26975023624364733927794525996, −3.67493208538793214417483424997, −2.68610735279515919365131723282, −1.58943084195169391148044960545, −0.843958229069094735950754984481, 0.843958229069094735950754984481, 1.58943084195169391148044960545, 2.68610735279515919365131723282, 3.67493208538793214417483424997, 4.26975023624364733927794525996, 5.37825277026734998627498746353, 6.34812805336864913009556585870, 6.49453266589921559642006169883, 7.70302577369861880861944948021, 7.914860535659067241209585074300

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