Properties

Label 2-7800-1.1-c1-0-12
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 − 0.647·7-s + 9-s + 0.502·11-s − 13-s + 4.73·17-s − 6.58·19-s + 0.647·21-s − 2.85·23-s − 27-s + 0.294·29-s + 2.20·31-s − 0.502·33-s − 7.28·37-s + 39-s + 7.58·41-s − 0.444·43-s − 8.07·47-s − 6.58·49-s − 4.73·51-s + 12.7·53-s + 6.58·57-s + 10.8·59-s + 11.4·61-s − 0.647·63-s + 8.63·67-s + 2.85·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.244·7-s + 0.333·9-s + 0.151·11-s − 0.277·13-s + 1.14·17-s − 1.50·19-s + 0.141·21-s − 0.594·23-s − 0.192·27-s + 0.0547·29-s + 0.395·31-s − 0.0874·33-s − 1.19·37-s + 0.160·39-s + 1.18·41-s − 0.0678·43-s − 1.17·47-s − 0.940·49-s − 0.662·51-s + 1.74·53-s + 0.871·57-s + 1.40·59-s + 1.46·61-s − 0.0815·63-s + 1.05·67-s + 0.343·69-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

\(L(1)\) \(\approx\) \(1.261190774\)
\(L(\frac12)\) \(\approx\) \(1.261190774\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.647T + 7T^{2} \)
11 \( 1 - 0.502T + 11T^{2} \)
17 \( 1 - 4.73T + 17T^{2} \)
19 \( 1 + 6.58T + 19T^{2} \)
23 \( 1 + 2.85T + 23T^{2} \)
29 \( 1 - 0.294T + 29T^{2} \)
31 \( 1 - 2.20T + 31T^{2} \)
37 \( 1 + 7.28T + 37T^{2} \)
41 \( 1 - 7.58T + 41T^{2} \)
43 \( 1 + 0.444T + 43T^{2} \)
47 \( 1 + 8.07T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 8.63T + 67T^{2} \)
71 \( 1 + 2.58T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 5.28T + 79T^{2} \)
83 \( 1 + 5.35T + 83T^{2} \)
89 \( 1 - 4.99T + 89T^{2} \)
97 \( 1 - 1.29T + 97T^{2} \)
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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

−7.905227614268081570523825754377, −6.96159134209910514324337612419, −6.54461912516276486204123509505, −5.72174318925472621599728182947, −5.19638382642675835643222051274, −4.25789914645874389154547687136, −3.68573879355457138829176699388, −2.63262926003324853705069689580, −1.72466624129393740727678591897, −0.57463280764360192965629254333, 0.57463280764360192965629254333, 1.72466624129393740727678591897, 2.63262926003324853705069689580, 3.68573879355457138829176699388, 4.25789914645874389154547687136, 5.19638382642675835643222051274, 5.72174318925472621599728182947, 6.54461912516276486204123509505, 6.96159134209910514324337612419, 7.905227614268081570523825754377

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