Properties

Label 2-7800-1.1-c1-0-1
Degree $2$
Conductor $7800$
Sign $1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4.48·7-s + 9-s − 1.20·11-s + 13-s − 3·17-s − 7.69·19-s + 4.48·21-s − 5.69·23-s − 27-s − 7.41·29-s − 1.20·31-s + 1.20·33-s + 0.719·37-s − 39-s + 1.28·41-s − 1.28·43-s − 5.92·47-s + 13.1·49-s + 3·51-s − 6.69·53-s + 7.69·57-s + 5.92·59-s + 9.41·61-s − 4.48·63-s − 11.2·67-s + 5.69·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.69·7-s + 0.333·9-s − 0.363·11-s + 0.277·13-s − 0.727·17-s − 1.76·19-s + 0.979·21-s − 1.18·23-s − 0.192·27-s − 1.37·29-s − 0.216·31-s + 0.209·33-s + 0.118·37-s − 0.160·39-s + 0.199·41-s − 0.195·43-s − 0.864·47-s + 1.87·49-s + 0.420·51-s − 0.919·53-s + 1.01·57-s + 0.771·59-s + 1.20·61-s − 0.565·63-s − 1.36·67-s + 0.685·69-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: no
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2446579303\)
\(L(\frac12)\) \(\approx\) \(0.2446579303\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.48T + 7T^{2} \)
11 \( 1 + 1.20T + 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 7.69T + 19T^{2} \)
23 \( 1 + 5.69T + 23T^{2} \)
29 \( 1 + 7.41T + 29T^{2} \)
31 \( 1 + 1.20T + 31T^{2} \)
37 \( 1 - 0.719T + 37T^{2} \)
41 \( 1 - 1.28T + 41T^{2} \)
43 \( 1 + 1.28T + 43T^{2} \)
47 \( 1 + 5.92T + 47T^{2} \)
53 \( 1 + 6.69T + 53T^{2} \)
59 \( 1 - 5.92T + 59T^{2} \)
61 \( 1 - 9.41T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 5.13T + 71T^{2} \)
73 \( 1 + 5.13T + 73T^{2} \)
79 \( 1 - 2.71T + 79T^{2} \)
83 \( 1 + 4.89T + 83T^{2} \)
89 \( 1 + 0.412T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−7.76130401350150217600924698937, −6.96888163212877470632315285502, −6.28151000040561662183816427258, −6.08960533366683173105032132290, −5.14705108993423152160153539752, −4.13559238269110737022208862734, −3.70423931910554467302953899005, −2.65589169370230981187220115398, −1.84295891356769524160335010379, −0.23763168854140627782972561740, 0.23763168854140627782972561740, 1.84295891356769524160335010379, 2.65589169370230981187220115398, 3.70423931910554467302953899005, 4.13559238269110737022208862734, 5.14705108993423152160153539752, 6.08960533366683173105032132290, 6.28151000040561662183816427258, 6.96888163212877470632315285502, 7.76130401350150217600924698937

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