Properties

Label 2-31200-1.1-c1-0-25
Degree $2$
Conductor $31200$
Sign $1$
Analytic cond. $249.133$
Root an. cond. $15.7839$
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  + 3-s + 7-s + 9-s + 3·11-s + 13-s + 5·17-s + 2·19-s + 21-s − 3·23-s + 27-s + 4·31-s + 3·33-s − 37-s + 39-s + 9·41-s + 2·43-s + 8·47-s − 6·49-s + 5·51-s + 53-s + 2·57-s − 4·59-s − 3·61-s + 63-s − 16·67-s − 3·69-s + 15·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 1.21·17-s + 0.458·19-s + 0.218·21-s − 0.625·23-s + 0.192·27-s + 0.718·31-s + 0.522·33-s − 0.164·37-s + 0.160·39-s + 1.40·41-s + 0.304·43-s + 1.16·47-s − 6/7·49-s + 0.700·51-s + 0.137·53-s + 0.264·57-s − 0.520·59-s − 0.384·61-s + 0.125·63-s − 1.95·67-s − 0.361·69-s + 1.78·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31200\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(249.133\)
Root analytic conductor: \(15.7839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{31200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.108112413\)
\(L(\frac12)\) \(\approx\) \(4.108112413\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−15.07003362063532, −14.39889288442260, −13.99152520384711, −13.87186574995310, −12.93034889397283, −12.40832803972086, −11.96089140818085, −11.44823854459163, −10.74822809045591, −10.25763074262898, −9.485996261190734, −9.297320628692162, −8.553663063451395, −7.870964362377594, −7.676542566683048, −6.850591455966797, −6.214267495868331, −5.666035266985298, −4.915513275297473, −4.203339152606012, −3.684210459801626, −3.032544723789980, −2.253222542018293, −1.420601337224999, −0.8200453084962188, 0.8200453084962188, 1.420601337224999, 2.253222542018293, 3.032544723789980, 3.684210459801626, 4.203339152606012, 4.915513275297473, 5.666035266985298, 6.214267495868331, 6.850591455966797, 7.676542566683048, 7.870964362377594, 8.553663063451395, 9.297320628692162, 9.485996261190734, 10.25763074262898, 10.74822809045591, 11.44823854459163, 11.96089140818085, 12.40832803972086, 12.93034889397283, 13.87186574995310, 13.99152520384711, 14.39889288442260, 15.07003362063532

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