Properties

Label 2-7800-1.1-c1-0-25
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 + 9-s − 4·11-s − 13-s − 8·17-s + 5·19-s + 4·23-s + 27-s + 9·29-s − 4·31-s − 4·33-s + 3·37-s − 39-s + 5·41-s − 6·43-s + 5·47-s − 7·49-s − 8·51-s + 5·53-s + 5·57-s − 6·59-s + 4·61-s + 3·67-s + 4·69-s + 7·71-s − 4·73-s − 79-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 1.94·17-s + 1.14·19-s + 0.834·23-s + 0.192·27-s + 1.67·29-s − 0.718·31-s − 0.696·33-s + 0.493·37-s − 0.160·39-s + 0.780·41-s − 0.914·43-s + 0.729·47-s − 49-s − 1.12·51-s + 0.686·53-s + 0.662·57-s − 0.781·59-s + 0.512·61-s + 0.366·67-s + 0.481·69-s + 0.830·71-s − 0.468·73-s − 0.112·79-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

\(L(1)\) \(\approx\) \(2.094160526\)
\(L(\frac12)\) \(\approx\) \(2.094160526\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−7.903131603519908289747785665839, −7.17993141418452131960325120897, −6.67617368485630854605762706128, −5.71600405991886094064258589832, −4.87786463283473746888550723725, −4.46324289568589546307923275372, −3.32051839779921621955058449345, −2.69514179703195495755979593252, −2.00064156841136673098677415870, −0.67905688090664365485868193436, 0.67905688090664365485868193436, 2.00064156841136673098677415870, 2.69514179703195495755979593252, 3.32051839779921621955058449345, 4.46324289568589546307923275372, 4.87786463283473746888550723725, 5.71600405991886094064258589832, 6.67617368485630854605762706128, 7.17993141418452131960325120897, 7.903131603519908289747785665839

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