Properties

Label 2-7800-1.1-c1-0-18
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 + 9-s − 4·11-s − 13-s − 2·17-s − 4·19-s − 8·23-s + 27-s + 6·29-s + 8·31-s − 4·33-s − 6·37-s − 39-s + 2·41-s + 12·43-s + 8·47-s − 7·49-s − 2·51-s + 2·53-s − 4·57-s + 12·59-s − 2·61-s − 12·67-s − 8·69-s + 16·71-s − 10·73-s + 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s − 0.160·39-s + 0.312·41-s + 1.82·43-s + 1.16·47-s − 49-s − 0.280·51-s + 0.274·53-s − 0.529·57-s + 1.56·59-s − 0.256·61-s − 1.46·67-s − 0.963·69-s + 1.89·71-s − 1.17·73-s + 0.900·79-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

\(L(1)\) \(\approx\) \(1.917819670\)
\(L(\frac12)\) \(\approx\) \(1.917819670\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.939428236970305679916521566968, −7.33133212186291559292656053100, −6.43524788089026139039377475493, −5.88592241444235658101323561619, −4.86565535063385090482663143888, −4.36054460607134966378609243541, −3.48486855853978324398418219304, −2.46066849465400339563544193095, −2.15821041599584095280752391820, −0.63876466895030982743652561062, 0.63876466895030982743652561062, 2.15821041599584095280752391820, 2.46066849465400339563544193095, 3.48486855853978324398418219304, 4.36054460607134966378609243541, 4.86565535063385090482663143888, 5.88592241444235658101323561619, 6.43524788089026139039377475493, 7.33133212186291559292656053100, 7.939428236970305679916521566968

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