Properties

Label 2-7800-1.1-c1-0-80
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.86·7-s + 9-s + 0.675·11-s − 13-s − 5.31·17-s − 0.806·19-s − 1.86·21-s + 8.73·23-s − 27-s − 3.96·29-s − 1.71·31-s − 0.675·33-s + 7.38·37-s + 39-s − 8.54·41-s − 10.4·43-s − 2.13·47-s − 3.50·49-s + 5.31·51-s + 6.19·53-s + 0.806·57-s + 14.4·59-s − 10.3·61-s + 1.86·63-s − 1.06·67-s − 8.73·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.706·7-s + 0.333·9-s + 0.203·11-s − 0.277·13-s − 1.28·17-s − 0.184·19-s − 0.407·21-s + 1.82·23-s − 0.192·27-s − 0.735·29-s − 0.307·31-s − 0.117·33-s + 1.21·37-s + 0.160·39-s − 1.33·41-s − 1.59·43-s − 0.310·47-s − 0.500·49-s + 0.743·51-s + 0.850·53-s + 0.106·57-s + 1.88·59-s − 1.32·61-s + 0.235·63-s − 0.129·67-s − 1.05·69-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: \(1\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.86T + 7T^{2} \)
11 \( 1 - 0.675T + 11T^{2} \)
17 \( 1 + 5.31T + 17T^{2} \)
19 \( 1 + 0.806T + 19T^{2} \)
23 \( 1 - 8.73T + 23T^{2} \)
29 \( 1 + 3.96T + 29T^{2} \)
31 \( 1 + 1.71T + 31T^{2} \)
37 \( 1 - 7.38T + 37T^{2} \)
41 \( 1 + 8.54T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 2.13T + 47T^{2} \)
53 \( 1 - 6.19T + 53T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 1.06T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 8.34T + 73T^{2} \)
79 \( 1 + 6.93T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 0.261T + 97T^{2} \)
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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

−7.30123541572636076832801252673, −6.88427270806825451833729780774, −6.17149092093856526155808040517, −5.24243689373481727927031955730, −4.82192485556139645333758822505, −4.10395983870123998653375327020, −3.10859223609433760355684432072, −2.10289925661687989317292007330, −1.26427182328738732777850530892, 0, 1.26427182328738732777850530892, 2.10289925661687989317292007330, 3.10859223609433760355684432072, 4.10395983870123998653375327020, 4.82192485556139645333758822505, 5.24243689373481727927031955730, 6.17149092093856526155808040517, 6.88427270806825451833729780774, 7.30123541572636076832801252673

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