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Properties

Label 2-979-979.21-c0-0-0
Degree $2$
Conductor $979$
Sign $-0.649 + 0.760i$
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  + (−0.415 + 0.0903i)3-s + (0.841 − 0.540i)4-s + (−1.27 − 1.10i)5-s + (−0.745 + 0.340i)9-s + (−0.654 − 0.755i)11-s + (−0.300 + 0.300i)12-s + (0.627 + 0.342i)15-s + (0.415 − 0.909i)16-s + (−1.66 − 0.239i)20-s + (−1.86 − 0.697i)23-s + (0.260 + 1.81i)25-s + (0.619 − 0.463i)27-s + (0.898 − 0.334i)31-s + (0.340 + 0.254i)33-s + (−0.442 + 0.689i)36-s + (−1.38 − 1.38i)37-s + ⋯
L(s)  = 1  + (−0.415 + 0.0903i)3-s + (0.841 − 0.540i)4-s + (−1.27 − 1.10i)5-s + (−0.745 + 0.340i)9-s + (−0.654 − 0.755i)11-s + (−0.300 + 0.300i)12-s + (0.627 + 0.342i)15-s + (0.415 − 0.909i)16-s + (−1.66 − 0.239i)20-s + (−1.86 − 0.697i)23-s + (0.260 + 1.81i)25-s + (0.619 − 0.463i)27-s + (0.898 − 0.334i)31-s + (0.340 + 0.254i)33-s + (−0.442 + 0.689i)36-s + (−1.38 − 1.38i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(979\)    =    \(11 \cdot 89\)
Sign: $-0.649 + 0.760i$
Analytic conductor: \(0.488584\)
Root analytic conductor: \(0.698988\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{979} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 979,\ (\ :0),\ -0.649 + 0.760i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5808481184\)
\(L(\frac12)\) \(\approx\) \(0.5808481184\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
good2 \( 1 + (-0.841 + 0.540i)T^{2} \)
3 \( 1 + (0.415 - 0.0903i)T + (0.909 - 0.415i)T^{2} \)
5 \( 1 + (1.27 + 1.10i)T + (0.142 + 0.989i)T^{2} \)
7 \( 1 + (0.989 - 0.142i)T^{2} \)
13 \( 1 + (0.909 - 0.415i)T^{2} \)
17 \( 1 + (0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.755 + 0.654i)T^{2} \)
23 \( 1 + (1.86 + 0.697i)T + (0.755 + 0.654i)T^{2} \)
29 \( 1 + (-0.989 + 0.142i)T^{2} \)
31 \( 1 + (-0.898 + 0.334i)T + (0.755 - 0.654i)T^{2} \)
37 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
41 \( 1 + (0.909 + 0.415i)T^{2} \)
43 \( 1 + (-0.989 - 0.142i)T^{2} \)
47 \( 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2} \)
53 \( 1 + (-0.708 + 1.10i)T + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (-1.83 - 0.398i)T + (0.909 + 0.415i)T^{2} \)
61 \( 1 + (0.281 - 0.959i)T^{2} \)
67 \( 1 + (0.474 + 0.304i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-1.45 + 1.25i)T + (0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.540 + 0.841i)T^{2} \)
97 \( 1 + (-0.989 + 1.14i)T + (-0.142 - 0.989i)T^{2} \)
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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

−10.15909836647978957558527771337, −8.876896889746122839832726162601, −8.143207688512082579457523565799, −7.60546302284848336046845140470, −6.30196754532829317000683379378, −5.56187611837839208783127126363, −4.77765835738255525872800342437, −3.64460888624384896081246865626, −2.31356689171269121995816211894, −0.53971717709319661149442619472, 2.30373759594896768913227556557, 3.25638707315663246621027119296, 4.02521267256727396398345591416, 5.46603519902056429738774384626, 6.58191222396925831811397506745, 7.04229367581401120597202539136, 7.937986259638820935720619715164, 8.417645514501517126329517884703, 10.07725241228551448304528474944, 10.58897762574744710453304450725

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