Properties

Label 2-7800-1.1-c1-0-36
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 − 7-s + 9-s + 3·11-s + 13-s + 7·17-s + 21-s + 7·23-s − 27-s − 4·29-s + 8·31-s − 3·33-s + 5·37-s − 39-s − 3·41-s + 8·43-s − 6·47-s − 6·49-s − 7·51-s + 11·53-s − 4·59-s + 61-s − 63-s − 12·67-s − 7·69-s − 9·71-s − 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 1.69·17-s + 0.218·21-s + 1.45·23-s − 0.192·27-s − 0.742·29-s + 1.43·31-s − 0.522·33-s + 0.821·37-s − 0.160·39-s − 0.468·41-s + 1.21·43-s − 0.875·47-s − 6/7·49-s − 0.980·51-s + 1.51·53-s − 0.520·59-s + 0.128·61-s − 0.125·63-s − 1.46·67-s − 0.842·69-s − 1.06·71-s − 0.702·73-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

\(L(1)\) \(\approx\) \(1.959145417\)
\(L(\frac12)\) \(\approx\) \(1.959145417\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−7.62494262271871471811213651890, −7.22126377339030391471896029577, −6.27406823965951843810127734362, −5.95675226497353076150822307162, −5.07792878991517465701598807997, −4.36622394107016148430873999216, −3.49128350408440612000270717550, −2.86202430318556327889806168318, −1.48050800838502808648840790286, −0.795922767386550153827753778796, 0.795922767386550153827753778796, 1.48050800838502808648840790286, 2.86202430318556327889806168318, 3.49128350408440612000270717550, 4.36622394107016148430873999216, 5.07792878991517465701598807997, 5.95675226497353076150822307162, 6.27406823965951843810127734362, 7.22126377339030391471896029577, 7.62494262271871471811213651890

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