Properties

Label 2-31200-1.1-c1-0-18
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 + 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: \(0\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.544537820\)
\(L(\frac12)\) \(\approx\) \(2.544537820\)
\(L(\frac{3}{2})\) 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

−15.17921961560862, −14.45473761669562, −14.13131165163724, −13.53498410006579, −13.01480952635478, −12.12591600848690, −11.94364803754922, −11.47554694947130, −10.76405367410276, −10.36348738413076, −9.947246756273973, −9.065725445912415, −8.390307476236617, −8.106476379345812, −7.296795658763661, −7.044060848327740, −5.939763394194235, −5.689826436355988, −4.855543454062475, −4.565589591644412, −3.908384793248021, −2.742579096913880, −2.303196884584353, −1.284410597883130, −0.7037483478066092, 0.7037483478066092, 1.284410597883130, 2.303196884584353, 2.742579096913880, 3.908384793248021, 4.565589591644412, 4.855543454062475, 5.689826436355988, 5.939763394194235, 7.044060848327740, 7.296795658763661, 8.106476379345812, 8.390307476236617, 9.065725445912415, 9.947246756273973, 10.36348738413076, 10.76405367410276, 11.47554694947130, 11.94364803754922, 12.12591600848690, 13.01480952635478, 13.53498410006579, 14.13131165163724, 14.45473761669562, 15.17921961560862

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