Properties

Label 2-31200-1.1-c1-0-36
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 − 4·7-s + 9-s + 2·11-s − 13-s − 2·19-s − 4·21-s + 2·23-s + 27-s + 10·29-s − 4·31-s + 2·33-s + 6·37-s − 39-s − 6·41-s − 8·43-s − 12·47-s + 9·49-s + 14·53-s − 2·57-s − 6·59-s + 2·61-s − 4·63-s + 4·67-s + 2·69-s − 14·73-s − 8·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.458·19-s − 0.872·21-s + 0.417·23-s + 0.192·27-s + 1.85·29-s − 0.718·31-s + 0.348·33-s + 0.986·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s + 1.92·53-s − 0.264·57-s − 0.781·59-s + 0.256·61-s − 0.503·63-s + 0.488·67-s + 0.240·69-s − 1.63·73-s − 0.911·77-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.20059927220424, −14.90743858658200, −14.34768259421058, −13.64714520236382, −13.24373066576512, −12.86693670864669, −12.18230956977830, −11.82535467450847, −11.06614070037236, −10.19176235529957, −10.06803218234404, −9.431035733912714, −8.866741367162902, −8.442656668397528, −7.722537422836265, −6.929327233541061, −6.615557094667764, −6.157400669568912, −5.251439340485169, −4.562039311834201, −3.867179406000895, −3.239156146159272, −2.807859777441858, −1.967761772739035, −1.008650305554277, 0, 1.008650305554277, 1.967761772739035, 2.807859777441858, 3.239156146159272, 3.867179406000895, 4.562039311834201, 5.251439340485169, 6.157400669568912, 6.615557094667764, 6.929327233541061, 7.722537422836265, 8.442656668397528, 8.866741367162902, 9.431035733912714, 10.06803218234404, 10.19176235529957, 11.06614070037236, 11.82535467450847, 12.18230956977830, 12.86693670864669, 13.24373066576512, 13.64714520236382, 14.34768259421058, 14.90743858658200, 15.20059927220424

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