Properties

Label 2-5850-1.1-c1-0-54
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 − 4·7-s − 8-s + 13-s + 4·14-s + 16-s − 2·17-s + 4·19-s + 8·23-s − 26-s − 4·28-s − 2·29-s − 8·31-s − 32-s + 2·34-s − 2·37-s − 4·38-s + 6·41-s − 12·43-s − 8·46-s + 9·49-s + 52-s + 10·53-s + 4·56-s + 2·58-s − 10·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 0.196·26-s − 0.755·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.648·38-s + 0.937·41-s − 1.82·43-s − 1.17·46-s + 9/7·49-s + 0.138·52-s + 1.37·53-s + 0.534·56-s + 0.262·58-s − 1.28·61-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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.66989657632459004812788027657, −6.99710497747098715630132652164, −6.59349388023917473502623632939, −5.73084052598874245562033811102, −5.02052022396818729106165437895, −3.70985819173759386781821472300, −3.23676801370547150675630240730, −2.33588772540446155513557717058, −1.11529347123600815773378494836, 0, 1.11529347123600815773378494836, 2.33588772540446155513557717058, 3.23676801370547150675630240730, 3.70985819173759386781821472300, 5.02052022396818729106165437895, 5.73084052598874245562033811102, 6.59349388023917473502623632939, 6.99710497747098715630132652164, 7.66989657632459004812788027657

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