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Properties

Label 2-979-979.340-c0-0-0
Degree $2$
Conductor $979$
Sign $0.394 - 0.918i$
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  + (−1.14 − 0.989i)3-s + (−0.654 − 0.755i)4-s + (−1.25 + 0.368i)5-s + (0.182 + 1.27i)9-s + (−0.959 − 0.281i)11-s + 1.51i·12-s + (1.80 + 0.822i)15-s + (−0.142 + 0.989i)16-s + (1.10 + 0.708i)20-s + (0.557 − 0.0801i)23-s + (0.601 − 0.386i)25-s + (0.232 − 0.361i)27-s + (1.80 + 0.258i)31-s + (0.817 + 1.27i)33-s + (0.841 − 0.970i)36-s + 1.51i·37-s + ⋯
L(s)  = 1  + (−1.14 − 0.989i)3-s + (−0.654 − 0.755i)4-s + (−1.25 + 0.368i)5-s + (0.182 + 1.27i)9-s + (−0.959 − 0.281i)11-s + 1.51i·12-s + (1.80 + 0.822i)15-s + (−0.142 + 0.989i)16-s + (1.10 + 0.708i)20-s + (0.557 − 0.0801i)23-s + (0.601 − 0.386i)25-s + (0.232 − 0.361i)27-s + (1.80 + 0.258i)31-s + (0.817 + 1.27i)33-s + (0.841 − 0.970i)36-s + 1.51i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(979\)    =    \(11 \cdot 89\)
Sign: $0.394 - 0.918i$
Analytic conductor: \(0.488584\)
Root analytic conductor: \(0.698988\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{979} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 979,\ (\ :0),\ 0.394 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1377458430\)
\(L(\frac12)\) \(\approx\) \(0.1377458430\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
good2 \( 1 + (0.654 + 0.755i)T^{2} \)
3 \( 1 + (1.14 + 0.989i)T + (0.142 + 0.989i)T^{2} \)
5 \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \)
7 \( 1 + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (-0.959 + 0.281i)T^{2} \)
23 \( 1 + (-0.557 + 0.0801i)T + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.841 - 0.540i)T^{2} \)
31 \( 1 + (-1.80 - 0.258i)T + (0.959 + 0.281i)T^{2} \)
37 \( 1 - 1.51iT - T^{2} \)
41 \( 1 + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
53 \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (1.49 - 1.29i)T + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (-0.544 + 0.627i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (-0.797 - 0.234i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2} \)
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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

−10.66788928826528143159176666066, −9.714201267605059726353575828504, −8.371322972773055005215956772779, −7.84749606829697472013658453224, −6.82859377224121331403943264734, −6.18515334973718047346633468982, −5.17843035671324976798019654489, −4.49285714420669246167480176373, −3.03442629542009182057075980189, −1.20682087555395120543503912222, 0.17739221541028969024196529171, 3.08762178298454200767755586632, 4.09291155719386607675221471460, 4.70341506249684813083817701230, 5.29687274011900328962674817969, 6.62710677182908390080973128510, 7.84263382065484790140490067684, 8.196658913157636310600771540523, 9.383267690893153721657634922811, 10.02168433136635965854996304449

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