Properties

Label 2-7800-1.1-c1-0-57
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 + 8·23-s − 27-s + 6·29-s − 4·31-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 + 12·59-s − 2·61-s + 4·63-s + 16·67-s − 8·69-s − 8·71-s + 6·73-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 + 1.66·23-s − 0.192·27-s + 1.11·29-s − 0.718·31-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 + 1.56·59-s − 0.256·61-s + 0.503·63-s + 1.95·67-s − 0.963·69-s − 0.949·71-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

\(L(1)\) \(\approx\) \(2.678130015\)
\(L(\frac12)\) \(\approx\) \(2.678130015\)
\(L(\frac{3}{2})\) 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 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−7.80542440039500531376663136604, −7.10877253024563994291573243713, −6.59576922989469601715608295698, −5.46440354768193225793215295972, −5.24394750464478131129333998303, −4.38940046318397123322095206448, −3.70057830196510452511665297389, −2.61936865045962813325527705580, −1.43290735649033519650331409341, −0.997249670085513809145283089468, 0.997249670085513809145283089468, 1.43290735649033519650331409341, 2.61936865045962813325527705580, 3.70057830196510452511665297389, 4.38940046318397123322095206448, 5.24394750464478131129333998303, 5.46440354768193225793215295972, 6.59576922989469601715608295698, 7.10877253024563994291573243713, 7.80542440039500531376663136604

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