Properties

Label 2-31200-1.1-c1-0-17
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 + 9-s + 4·11-s − 13-s + 6·17-s − 8·19-s − 8·23-s + 27-s + 2·29-s + 8·31-s + 4·33-s + 10·37-s − 39-s + 6·41-s + 4·43-s − 7·49-s + 6·51-s + 14·53-s − 8·57-s − 12·59-s − 10·61-s + 8·67-s − 8·69-s + 14·73-s − 4·79-s + 81-s − 4·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 1.45·17-s − 1.83·19-s − 1.66·23-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.696·33-s + 1.64·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s − 49-s + 0.840·51-s + 1.92·53-s − 1.05·57-s − 1.56·59-s − 1.28·61-s + 0.977·67-s − 0.963·69-s + 1.63·73-s − 0.450·79-s + 1/9·81-s − 0.439·83-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

\(L(1)\) \(\approx\) \(3.275308206\)
\(L(\frac12)\) \(\approx\) \(3.275308206\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.08223008307911, −14.37640126495649, −14.21285904607213, −13.70190572916052, −12.88684660844855, −12.39177228226784, −12.07594983678306, −11.42344737168292, −10.75496698426824, −10.08878746015974, −9.736819432596318, −9.197932740722463, −8.461872194551660, −8.063267446260475, −7.589738771444614, −6.759627532199744, −6.154086210250447, −5.894436626442579, −4.744976155927658, −4.189366213004798, −3.836707343902308, −2.897913930808145, −2.306499064666468, −1.508738691607810, −0.6894980735857294, 0.6894980735857294, 1.508738691607810, 2.306499064666468, 2.897913930808145, 3.836707343902308, 4.189366213004798, 4.744976155927658, 5.894436626442579, 6.154086210250447, 6.759627532199744, 7.589738771444614, 8.063267446260475, 8.461872194551660, 9.197932740722463, 9.736819432596318, 10.08878746015974, 10.75496698426824, 11.42344737168292, 12.07594983678306, 12.39177228226784, 12.88684660844855, 13.70190572916052, 14.21285904607213, 14.37640126495649, 15.08223008307911

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