Properties

Label 2-31200-1.1-c1-0-20
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 + 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: \(0\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.750130760\)
\(L(\frac12)\) \(\approx\) \(3.750130760\)
\(L(\frac{3}{2})\) 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

−14.90841676893268, −14.63873625401329, −14.07427238755966, −13.60805034356749, −12.90944582411162, −12.66612727499276, −11.71600625993009, −11.49721758778714, −10.73813808924181, −10.34624813057107, −9.605016723276618, −9.000267764065365, −8.702948663411360, −7.981707162953913, −7.435988171451562, −7.013035698985750, −6.264473620571650, −5.520780999605049, −4.824835917464926, −4.455133612780830, −3.643676454542231, −2.838776356291617, −2.395801887997325, −1.444659945213209, −0.7607011079635210, 0.7607011079635210, 1.444659945213209, 2.395801887997325, 2.838776356291617, 3.643676454542231, 4.455133612780830, 4.824835917464926, 5.520780999605049, 6.264473620571650, 7.013035698985750, 7.435988171451562, 7.981707162953913, 8.702948663411360, 9.000267764065365, 9.605016723276618, 10.34624813057107, 10.73813808924181, 11.49721758778714, 11.71600625993009, 12.66612727499276, 12.90944582411162, 13.60805034356749, 14.07427238755966, 14.63873625401329, 14.90841676893268

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