Properties

Label 2-7800-1.1-c1-0-38
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 − 2·7-s + 9-s + 4·11-s + 13-s + 2·19-s − 2·21-s + 2·23-s + 27-s + 4·29-s + 4·31-s + 4·33-s + 2·37-s + 39-s − 6·41-s + 4·43-s − 8·47-s − 3·49-s + 2·53-s + 2·57-s + 4·59-s − 2·61-s − 2·63-s − 8·67-s + 2·69-s + 8·71-s + 4·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.458·19-s − 0.436·21-s + 0.417·23-s + 0.192·27-s + 0.742·29-s + 0.718·31-s + 0.696·33-s + 0.328·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.264·57-s + 0.520·59-s − 0.256·61-s − 0.251·63-s − 0.977·67-s + 0.240·69-s + 0.949·71-s + 0.468·73-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

\(L(1)\) \(\approx\) \(2.699401892\)
\(L(\frac12)\) \(\approx\) \(2.699401892\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.970887846116897845337911764056, −7.00602061281196743622001027859, −6.60925303064095719357376751039, −5.94627902094419587493385407810, −4.93678056529372727717078038163, −4.17706540036201829301436535489, −3.41731248257339669585809020486, −2.88127452579023111506453617323, −1.76204806578458563268563154525, −0.825290408356212376382984222921, 0.825290408356212376382984222921, 1.76204806578458563268563154525, 2.88127452579023111506453617323, 3.41731248257339669585809020486, 4.17706540036201829301436535489, 4.93678056529372727717078038163, 5.94627902094419587493385407810, 6.60925303064095719357376751039, 7.00602061281196743622001027859, 7.970887846116897845337911764056

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