Properties

Label 2-7800-1.1-c1-0-0
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 − 2.60·7-s + 9-s − 5.61·11-s − 13-s − 5.77·17-s − 0.234·19-s + 2.60·21-s − 7.01·23-s − 27-s + 4.20·29-s + 4.41·31-s + 5.61·33-s + 2.96·37-s + 39-s − 10.9·41-s − 0.188·43-s − 7.84·47-s − 0.234·49-s + 5.77·51-s − 11.9·53-s + 0.234·57-s − 12.1·59-s − 2.98·61-s − 2.60·63-s − 3.56·67-s + 7.01·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.983·7-s + 0.333·9-s − 1.69·11-s − 0.277·13-s − 1.40·17-s − 0.0537·19-s + 0.567·21-s − 1.46·23-s − 0.192·27-s + 0.780·29-s + 0.792·31-s + 0.977·33-s + 0.487·37-s + 0.160·39-s − 1.71·41-s − 0.0287·43-s − 1.14·47-s − 0.0334·49-s + 0.809·51-s − 1.64·53-s + 0.0310·57-s − 1.58·59-s − 0.381·61-s − 0.327·63-s − 0.436·67-s + 0.844·69-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

\(L(1)\) \(\approx\) \(0.2447124801\)
\(L(\frac12)\) \(\approx\) \(0.2447124801\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.60T + 7T^{2} \)
11 \( 1 + 5.61T + 11T^{2} \)
17 \( 1 + 5.77T + 17T^{2} \)
19 \( 1 + 0.234T + 19T^{2} \)
23 \( 1 + 7.01T + 23T^{2} \)
29 \( 1 - 4.20T + 29T^{2} \)
31 \( 1 - 4.41T + 31T^{2} \)
37 \( 1 - 2.96T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 0.188T + 43T^{2} \)
47 \( 1 + 7.84T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + 2.98T + 61T^{2} \)
67 \( 1 + 3.56T + 67T^{2} \)
71 \( 1 - 3.76T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 + 4.96T + 79T^{2} \)
83 \( 1 + 3.39T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 5.20T + 97T^{2} \)
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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

−8.025218193494054407852521017344, −6.98017740107526147463197954916, −6.41504635879154705545131941540, −5.93176666458126626974634265198, −4.90855440244577630247411660590, −4.58950223956366971765316841324, −3.42830709720399805649472889230, −2.70053576957027628798496210121, −1.86302681645402985529313277070, −0.23631501577451695489503474969, 0.23631501577451695489503474969, 1.86302681645402985529313277070, 2.70053576957027628798496210121, 3.42830709720399805649472889230, 4.58950223956366971765316841324, 4.90855440244577630247411660590, 5.93176666458126626974634265198, 6.41504635879154705545131941540, 6.98017740107526147463197954916, 8.025218193494054407852521017344

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