Properties

Label 2-5850-1.1-c1-0-5
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 − 2·7-s − 8-s − 13-s + 2·14-s + 16-s + 2·19-s − 6·23-s + 26-s − 2·28-s + 8·31-s − 32-s − 2·37-s − 2·38-s − 6·41-s + 4·43-s + 6·46-s − 3·49-s − 52-s − 6·53-s + 2·56-s + 14·61-s − 8·62-s + 64-s + 4·67-s + 4·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.458·19-s − 1.25·23-s + 0.196·26-s − 0.377·28-s + 1.43·31-s − 0.176·32-s − 0.328·37-s − 0.324·38-s − 0.937·41-s + 0.609·43-s + 0.884·46-s − 3/7·49-s − 0.138·52-s − 0.824·53-s + 0.267·56-s + 1.79·61-s − 1.01·62-s + 1/8·64-s + 0.488·67-s + 0.468·73-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

\(L(1)\) \(\approx\) \(0.9584057276\)
\(L(\frac12)\) \(\approx\) \(0.9584057276\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.181916686992815231690366948107, −7.45683248093908224431285467559, −6.72529053692362922426440322752, −6.17142689234210538228627385739, −5.37908738825114272050743174154, −4.40710216635884316616874198496, −3.47995811049370451046055776313, −2.72574292422826359482229792049, −1.78178789303379312995380753176, −0.56489928294757840379112714493, 0.56489928294757840379112714493, 1.78178789303379312995380753176, 2.72574292422826359482229792049, 3.47995811049370451046055776313, 4.40710216635884316616874198496, 5.37908738825114272050743174154, 6.17142689234210538228627385739, 6.72529053692362922426440322752, 7.45683248093908224431285467559, 8.181916686992815231690366948107

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