Properties

Label 2-980-1.1-c1-0-8
Degree $2$
Conductor $980$
Sign $1$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
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  + 3·3-s + 5-s + 6·9-s − 2·11-s + 6·13-s + 3·15-s − 2·17-s − 9·23-s + 25-s + 9·27-s + 3·29-s − 2·31-s − 6·33-s + 8·37-s + 18·39-s − 5·41-s + 43-s + 6·45-s − 8·47-s − 6·51-s + 4·53-s − 2·55-s + 8·59-s − 7·61-s + 6·65-s − 3·67-s − 27·69-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 2·9-s − 0.603·11-s + 1.66·13-s + 0.774·15-s − 0.485·17-s − 1.87·23-s + 1/5·25-s + 1.73·27-s + 0.557·29-s − 0.359·31-s − 1.04·33-s + 1.31·37-s + 2.88·39-s − 0.780·41-s + 0.152·43-s + 0.894·45-s − 1.16·47-s − 0.840·51-s + 0.549·53-s − 0.269·55-s + 1.04·59-s − 0.896·61-s + 0.744·65-s − 0.366·67-s − 3.25·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{980} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.133303692\)
\(L(\frac12)\) \(\approx\) \(3.133303692\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.861341895886835928479556830898, −9.035811175689496960229121791675, −8.323416847465527228736051049638, −7.88368303014610380901450609320, −6.70298540851638126059873615067, −5.80575855175278089882628697130, −4.36419093915488757070038212385, −3.56036060499126781046326462835, −2.56199961749627251052556531480, −1.60608326000150852234991263675, 1.60608326000150852234991263675, 2.56199961749627251052556531480, 3.56036060499126781046326462835, 4.36419093915488757070038212385, 5.80575855175278089882628697130, 6.70298540851638126059873615067, 7.88368303014610380901450609320, 8.323416847465527228736051049638, 9.035811175689496960229121791675, 9.861341895886835928479556830898

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