Properties

Label 2-31200-1.1-c1-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s + 9-s − 13-s − 2·17-s − 2·19-s + 2·21-s − 8·23-s − 27-s + 6·29-s + 2·31-s + 6·37-s + 39-s − 4·43-s − 8·47-s − 3·49-s + 2·51-s + 6·53-s + 2·57-s + 4·59-s + 2·61-s − 2·63-s − 2·67-s + 8·69-s + 4·71-s + 2·73-s + 12·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.277·13-s − 0.485·17-s − 0.458·19-s + 0.436·21-s − 1.66·23-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.986·37-s + 0.160·39-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 0.264·57-s + 0.520·59-s + 0.256·61-s − 0.251·63-s − 0.244·67-s + 0.963·69-s + 0.474·71-s + 0.234·73-s + 1.35·79-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: \(1\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−15.43261381377735, −14.86862016652658, −14.32189973310468, −13.63524572917577, −13.22315586777419, −12.69322512777468, −12.09434136078966, −11.74695247385399, −11.14462386936097, −10.36469186539355, −10.12782359084807, −9.542725189872137, −8.945575067030264, −8.139149088527018, −7.826938784250187, −6.900805140072807, −6.336683341201082, −6.218694728993037, −5.253045514472212, −4.709688646877307, −4.038794486698128, −3.426219401952989, −2.530376004952732, −1.934958098430834, −0.8182746791604373, 0, 0.8182746791604373, 1.934958098430834, 2.530376004952732, 3.426219401952989, 4.038794486698128, 4.709688646877307, 5.253045514472212, 6.218694728993037, 6.336683341201082, 6.900805140072807, 7.826938784250187, 8.139149088527018, 8.945575067030264, 9.542725189872137, 10.12782359084807, 10.36469186539355, 11.14462386936097, 11.74695247385399, 12.09434136078966, 12.69322512777468, 13.22315586777419, 13.63524572917577, 14.32189973310468, 14.86862016652658, 15.43261381377735

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