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Properties

Label 2-980-5.4-c1-0-13
Degree $2$
Conductor $980$
Sign $0.955 - 0.293i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s + (2.13 − 0.656i)5-s + 2.27·11-s − 6.09i·13-s + (1.13 + 3.70i)15-s − 4.77i·17-s + 4.27·19-s + 0.894i·23-s + (4.13 − 2.80i)25-s + 5.19i·27-s + 3.27·29-s − 4.27·31-s + 3.94i·33-s + 5.61i·37-s + 10.5·39-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.955 − 0.293i)5-s + 0.685·11-s − 1.68i·13-s + (0.293 + 0.955i)15-s − 1.15i·17-s + 0.980·19-s + 0.186i·23-s + (0.827 − 0.561i)25-s + 1.00i·27-s + 0.608·29-s − 0.767·31-s + 0.685i·33-s + 0.923i·37-s + 1.68·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.955 - 0.293i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.955 - 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04021 + 0.306350i\)
\(L(\frac12)\) \(\approx\) \(2.04021 + 0.306350i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 + 0.656i)T \)
7 \( 1 \)
good3 \( 1 - 1.73iT - 3T^{2} \)
11 \( 1 - 2.27T + 11T^{2} \)
13 \( 1 + 6.09iT - 13T^{2} \)
17 \( 1 + 4.77iT - 17T^{2} \)
19 \( 1 - 4.27T + 19T^{2} \)
23 \( 1 - 0.894iT - 23T^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 + 4.27T + 31T^{2} \)
37 \( 1 - 5.61iT - 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 6.50iT - 43T^{2} \)
47 \( 1 + 2.15iT - 47T^{2} \)
53 \( 1 - 7.40iT - 53T^{2} \)
59 \( 1 + 4.27T + 59T^{2} \)
61 \( 1 - 1.54T + 61T^{2} \)
67 \( 1 - 13.9iT - 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 2.15iT - 73T^{2} \)
79 \( 1 - 0.274T + 79T^{2} \)
83 \( 1 - 5.67iT - 83T^{2} \)
89 \( 1 - 7T + 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−10.06034024267647370788569403097, −9.376549480145638012281000144870, −8.679168755041749053163975472392, −7.53312433866475062999696157685, −6.55189533247964646349836358030, −5.27761544728862968081407774533, −5.12262614404867285132776932416, −3.72775695241190780099359122473, −2.80284053534712485165337929509, −1.15338412196087249873991553514, 1.46707223415394666313660463887, 2.03410528293657867251889262524, 3.54953427983868203590556281641, 4.74155372741330535889366609832, 6.00934625557292535539049770195, 6.61569728695963710160373894604, 7.16053451874479466730082982188, 8.270127384898417845766557641650, 9.223899700719123256339317963550, 9.800054118687206859668603999647

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