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Properties

Label 2-979-979.692-c0-0-0
Degree $2$
Conductor $979$
Sign $0.969 + 0.246i$
Analytic cond. $0.488584$
Root an. cond. $0.698988$
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  + (−0.841 + 1.54i)3-s + (−0.959 − 0.281i)4-s + (−1.74 − 0.797i)5-s + (−1.12 − 1.75i)9-s + (0.415 + 0.909i)11-s + (1.24 − 1.24i)12-s + (2.69 − 2.01i)15-s + (0.841 + 0.540i)16-s + (1.45 + 1.25i)20-s + (0.398 − 1.83i)23-s + (1.75 + 2.02i)25-s + (1.89 − 0.135i)27-s + (−0.254 − 1.17i)31-s + (−1.75 − 0.125i)33-s + (0.586 + 1.99i)36-s + (0.677 + 0.677i)37-s + ⋯
L(s)  = 1  + (−0.841 + 1.54i)3-s + (−0.959 − 0.281i)4-s + (−1.74 − 0.797i)5-s + (−1.12 − 1.75i)9-s + (0.415 + 0.909i)11-s + (1.24 − 1.24i)12-s + (2.69 − 2.01i)15-s + (0.841 + 0.540i)16-s + (1.45 + 1.25i)20-s + (0.398 − 1.83i)23-s + (1.75 + 2.02i)25-s + (1.89 − 0.135i)27-s + (−0.254 − 1.17i)31-s + (−1.75 − 0.125i)33-s + (0.586 + 1.99i)36-s + (0.677 + 0.677i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(979\)    =    \(11 \cdot 89\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(0.488584\)
Root analytic conductor: \(0.698988\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{979} (692, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 979,\ (\ :0),\ 0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3301152235\)
\(L(\frac12)\) \(\approx\) \(0.3301152235\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
good2 \( 1 + (0.959 + 0.281i)T^{2} \)
3 \( 1 + (0.841 - 1.54i)T + (-0.540 - 0.841i)T^{2} \)
5 \( 1 + (1.74 + 0.797i)T + (0.654 + 0.755i)T^{2} \)
7 \( 1 + (0.755 - 0.654i)T^{2} \)
13 \( 1 + (-0.540 - 0.841i)T^{2} \)
17 \( 1 + (-0.959 + 0.281i)T^{2} \)
19 \( 1 + (-0.909 - 0.415i)T^{2} \)
23 \( 1 + (-0.398 + 1.83i)T + (-0.909 - 0.415i)T^{2} \)
29 \( 1 + (-0.755 + 0.654i)T^{2} \)
31 \( 1 + (0.254 + 1.17i)T + (-0.909 + 0.415i)T^{2} \)
37 \( 1 + (-0.677 - 0.677i)T + iT^{2} \)
41 \( 1 + (-0.540 + 0.841i)T^{2} \)
43 \( 1 + (-0.755 - 0.654i)T^{2} \)
47 \( 1 + (-0.425 + 1.45i)T + (-0.841 - 0.540i)T^{2} \)
53 \( 1 + (0.234 + 0.797i)T + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.203 + 0.373i)T + (-0.540 + 0.841i)T^{2} \)
61 \( 1 + (0.989 - 0.142i)T^{2} \)
67 \( 1 + (-1.89 + 0.557i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (0.258 - 0.118i)T + (0.654 - 0.755i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.415 + 0.909i)T^{2} \)
83 \( 1 + (-0.281 - 0.959i)T^{2} \)
97 \( 1 + (-0.755 + 1.65i)T + (-0.654 - 0.755i)T^{2} \)
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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

−10.09545968740179626485058104530, −9.409951173266059614887473175300, −8.699738590619888791936588119019, −7.983932733281864661525443009117, −6.64968842682910724054607412745, −5.36843505184852396055586161804, −4.57689309474676142638361346607, −4.34029618952224579567266008062, −3.52720816557373459396559505116, −0.49830730358581025940857390209, 0.987660240979724868060312152079, 3.05096907138557458701204238317, 3.87570199177675847790162206017, 5.10927889746319299710628339093, 6.12558070771579335604883593245, 7.07520804027022519168127067465, 7.65388559023256624579329010898, 8.196101810883713010967060767230, 9.128020038592376956997695369861, 10.64502814387380689370609809543

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