Properties

Label 2-5850-1.1-c1-0-66
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·7-s − 8-s − 2·11-s − 13-s − 2·14-s + 16-s − 2·17-s − 4·19-s + 2·22-s + 26-s + 2·28-s − 4·29-s + 8·31-s − 32-s + 2·34-s + 6·37-s + 4·38-s + 6·41-s + 4·43-s − 2·44-s + 8·47-s − 3·49-s − 52-s − 2·53-s − 2·56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 0.603·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.426·22-s + 0.196·26-s + 0.377·28-s − 0.742·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.986·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s − 0.301·44-s + 1.16·47-s − 3/7·49-s − 0.138·52-s − 0.274·53-s − 0.267·56-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: \(1\)
Selberg data: \((2,\ 5850,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−7.78519645517762820623028167123, −7.32838381905593018981120818305, −6.31832145025474633025184492985, −5.80169073208914330079009737256, −4.72281757205917831378339343510, −4.25193280472204900145168891729, −2.91436061343674150027675689504, −2.26515124923791727175324286091, −1.28114720206099037337893092765, 0, 1.28114720206099037337893092765, 2.26515124923791727175324286091, 2.91436061343674150027675689504, 4.25193280472204900145168891729, 4.72281757205917831378339343510, 5.80169073208914330079009737256, 6.31832145025474633025184492985, 7.32838381905593018981120818305, 7.78519645517762820623028167123

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