Properties

Label 2-7800-1.1-c1-0-16
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 − 4·7-s + 9-s + 4·11-s − 13-s − 6·17-s − 4·21-s + 4·23-s + 27-s − 6·29-s − 8·31-s + 4·33-s + 2·37-s − 39-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s − 6·51-s + 2·53-s + 4·59-s + 14·61-s − 4·63-s + 12·67-s + 4·69-s − 8·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.45·17-s − 0.872·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.840·51-s + 0.274·53-s + 0.520·59-s + 1.79·61-s − 0.503·63-s + 1.46·67-s + 0.481·69-s − 0.949·71-s + 1.17·73-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

\(L(1)\) \(\approx\) \(1.835570400\)
\(L(\frac12)\) \(\approx\) \(1.835570400\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−7.76019256755424498189251499443, −7.01496845223186047092345362595, −6.63453719195890554388236000232, −5.96463827067582150364417721508, −5.00230078087916899658888094512, −3.92383451095228480659126951134, −3.69772392055574236854917436117, −2.68861134755498697986579936170, −1.96347270948825970656484566127, −0.63481869045556656904141253025, 0.63481869045556656904141253025, 1.96347270948825970656484566127, 2.68861134755498697986579936170, 3.69772392055574236854917436117, 3.92383451095228480659126951134, 5.00230078087916899658888094512, 5.96463827067582150364417721508, 6.63453719195890554388236000232, 7.01496845223186047092345362595, 7.76019256755424498189251499443

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