Properties

Label 2-7800-1.1-c1-0-83
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5·7-s + 9-s + 5·11-s + 13-s − 3·17-s − 4·19-s − 5·21-s + 5·23-s + 27-s − 4·29-s + 5·33-s − 7·37-s + 39-s + 11·41-s − 12·43-s + 6·47-s + 18·49-s − 3·51-s + 53-s − 4·57-s + 12·59-s − 7·61-s − 5·63-s + 4·67-s + 5·69-s − 7·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.88·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.727·17-s − 0.917·19-s − 1.09·21-s + 1.04·23-s + 0.192·27-s − 0.742·29-s + 0.870·33-s − 1.15·37-s + 0.160·39-s + 1.71·41-s − 1.82·43-s + 0.875·47-s + 18/7·49-s − 0.420·51-s + 0.137·53-s − 0.529·57-s + 1.56·59-s − 0.896·61-s − 0.629·63-s + 0.488·67-s + 0.601·69-s − 0.830·71-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: \(1\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 5 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−7.20097359490771831425027617382, −6.85877261362368570323372411334, −6.32184353209160291653618942462, −5.61087398737224947947179430942, −4.34149699735162375887751579818, −3.83798698840120102530072050724, −3.19117005113441443575831995119, −2.39657922240394678704191084242, −1.29439956450967371194308707026, 0, 1.29439956450967371194308707026, 2.39657922240394678704191084242, 3.19117005113441443575831995119, 3.83798698840120102530072050724, 4.34149699735162375887751579818, 5.61087398737224947947179430942, 6.32184353209160291653618942462, 6.85877261362368570323372411334, 7.20097359490771831425027617382

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