Properties

Label 2-2400-1.1-c1-0-35
Degree $2$
Conductor $2400$
Sign $-1$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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 + 7-s + 9-s − 4·11-s − 3·13-s − 4·17-s − 19-s + 21-s + 27-s − 8·29-s + 31-s − 4·33-s + 2·37-s − 3·39-s + 2·41-s − 11·43-s − 2·47-s − 6·49-s − 4·51-s − 10·53-s − 57-s − 6·59-s + 11·61-s + 63-s + 9·67-s + 6·71-s − 14·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.832·13-s − 0.970·17-s − 0.229·19-s + 0.218·21-s + 0.192·27-s − 1.48·29-s + 0.179·31-s − 0.696·33-s + 0.328·37-s − 0.480·39-s + 0.312·41-s − 1.67·43-s − 0.291·47-s − 6/7·49-s − 0.560·51-s − 1.37·53-s − 0.132·57-s − 0.781·59-s + 1.40·61-s + 0.125·63-s + 1.09·67-s + 0.712·71-s − 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−8.430815690871999608253662507706, −7.908565769649762575481887809066, −7.21207563951617328702489162850, −6.34148953270907415258289494499, −5.20094162310079706511563061578, −4.69054360413855447420301695810, −3.59388695481838457702722715252, −2.58291128002808959360122509250, −1.84886089377979123884783673436, 0, 1.84886089377979123884783673436, 2.58291128002808959360122509250, 3.59388695481838457702722715252, 4.69054360413855447420301695810, 5.20094162310079706511563061578, 6.34148953270907415258289494499, 7.21207563951617328702489162850, 7.908565769649762575481887809066, 8.430815690871999608253662507706

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