Properties

Label 2-5850-1.1-c1-0-22
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 − 3·7-s + 8-s + 3·11-s − 13-s − 3·14-s + 16-s + 3·17-s + 3·22-s + 4·23-s − 26-s − 3·28-s − 5·29-s − 3·31-s + 32-s + 3·34-s + 12·37-s − 2·41-s − 4·43-s + 3·44-s + 4·46-s + 3·47-s + 2·49-s − 52-s + 9·53-s − 3·56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s + 0.904·11-s − 0.277·13-s − 0.801·14-s + 1/4·16-s + 0.727·17-s + 0.639·22-s + 0.834·23-s − 0.196·26-s − 0.566·28-s − 0.928·29-s − 0.538·31-s + 0.176·32-s + 0.514·34-s + 1.97·37-s − 0.312·41-s − 0.609·43-s + 0.452·44-s + 0.589·46-s + 0.437·47-s + 2/7·49-s − 0.138·52-s + 1.23·53-s − 0.400·56-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.919982644\)
\(L(\frac12)\) \(\approx\) \(2.919982644\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−7.892621110288652042488521953644, −7.22681572445195529290233725233, −6.56200767706577465420344296023, −6.00449593130245924585161109271, −5.27134711941384447648237394295, −4.37504655833147681714468745295, −3.58370175708924027388339437408, −3.07148446869622941697388812954, −2.03296075777599711950431073239, −0.813520562214492182119142558246, 0.813520562214492182119142558246, 2.03296075777599711950431073239, 3.07148446869622941697388812954, 3.58370175708924027388339437408, 4.37504655833147681714468745295, 5.27134711941384447648237394295, 6.00449593130245924585161109271, 6.56200767706577465420344296023, 7.22681572445195529290233725233, 7.892621110288652042488521953644

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