Properties

Label 2-7800-1.1-c1-0-69
Degree $2$
Conductor $7800$
Sign $1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
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 + 5·7-s + 9-s + 11-s − 13-s + 3·17-s + 6·19-s + 5·21-s + 7·23-s + 27-s + 6·29-s − 2·31-s + 33-s − 37-s − 39-s + 7·41-s − 8·43-s − 2·47-s + 18·49-s + 3·51-s − 13·53-s + 6·57-s − 8·59-s − 7·61-s + 5·63-s − 12·67-s + 7·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.88·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.727·17-s + 1.37·19-s + 1.09·21-s + 1.45·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.174·33-s − 0.164·37-s − 0.160·39-s + 1.09·41-s − 1.21·43-s − 0.291·47-s + 18/7·49-s + 0.420·51-s − 1.78·53-s + 0.794·57-s − 1.04·59-s − 0.896·61-s + 0.629·63-s − 1.46·67-s + 0.842·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(62.2833\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.962275131\)
\(L(\frac12)\) \(\approx\) \(3.962275131\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.83017535383499525146624427944, −7.45141318674944042941769132047, −6.62695754862225108798540262964, −5.55066952945700934449711951822, −4.92628380276933848337913877144, −4.50312809532376402513028618726, −3.39390598728163983004781714616, −2.74543013383357583921190572895, −1.59871287829168715448920885412, −1.12283094518913720550671210497, 1.12283094518913720550671210497, 1.59871287829168715448920885412, 2.74543013383357583921190572895, 3.39390598728163983004781714616, 4.50312809532376402513028618726, 4.92628380276933848337913877144, 5.55066952945700934449711951822, 6.62695754862225108798540262964, 7.45141318674944042941769132047, 7.83017535383499525146624427944

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