Properties

Label 2-980-980.799-c0-0-0
Degree $2$
Conductor $980$
Sign $0.801 + 0.598i$
Analytic cond. $0.489083$
Root an. cond. $0.699345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.777 + 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)5-s + (0.777 + 0.974i)6-s + (0.900 + 0.433i)7-s + (−0.623 + 0.781i)8-s + (−0.123 − 0.541i)9-s + (−0.623 − 0.781i)10-s + (1.12 − 0.541i)12-s + (0.623 − 0.781i)14-s + (0.277 + 1.21i)15-s + (0.623 + 0.781i)16-s − 0.554·18-s + (−0.900 + 0.433i)20-s + (−1.12 + 0.541i)21-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.777 + 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)5-s + (0.777 + 0.974i)6-s + (0.900 + 0.433i)7-s + (−0.623 + 0.781i)8-s + (−0.123 − 0.541i)9-s + (−0.623 − 0.781i)10-s + (1.12 − 0.541i)12-s + (0.623 − 0.781i)14-s + (0.277 + 1.21i)15-s + (0.623 + 0.781i)16-s − 0.554·18-s + (−0.900 + 0.433i)20-s + (−1.12 + 0.541i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.801 + 0.598i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.801 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9788126551\)
\(L(\frac12)\) \(\approx\) \(0.9788126551\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
good3 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.623 + 0.781i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.900 - 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.34530991924404823930953305060, −9.452155667858362759447104990376, −8.872306565612576551839565997798, −7.968528680854625806827841626455, −6.18666433641034223173661907239, −5.33408840265871537983279562309, −4.86252003664822354211519184532, −4.15529939304290375261098484025, −2.64405627866272526515290270100, −1.34097608348028343016065956745, 1.30851887438826301727517610117, 2.97369842387203172125456815135, 4.46578934803074059405648017685, 5.36912787626619798347866556839, 6.18522970786729600504910007904, 6.91241138038862862238823909852, 7.36184117479901622369655356246, 8.333835363881519608793803056839, 9.283468218780219889094901487094, 10.43424981927449387413532568064

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