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Properties

Label 2-980-140.139-c1-0-105
Degree $2$
Conductor $980$
Sign $-0.654 + 0.755i$
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  + 1.41·2-s − 2.80i·3-s + 2.00·4-s − 2.23i·5-s − 3.96i·6-s + 2.82·8-s − 4.87·9-s − 3.16i·10-s − 5.61i·12-s − 6.27·15-s + 4.00·16-s − 6.89·18-s − 4.47i·20-s + 8.92·23-s − 7.93i·24-s − 5.00·25-s + ⋯
L(s)  = 1  + 1.00·2-s − 1.61i·3-s + 1.00·4-s − 0.999i·5-s − 1.61i·6-s + 1.00·8-s − 1.62·9-s − 1.00i·10-s − 1.61i·12-s − 1.61·15-s + 1.00·16-s − 1.62·18-s − 1.00i·20-s + 1.86·23-s − 1.61i·24-s − 1.00·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23962 - 2.71342i\)
\(L(\frac12)\) \(\approx\) \(1.23962 - 2.71342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
5 \( 1 + 2.23iT \)
7 \( 1 \)
good3 \( 1 + 2.80iT - 3T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 8.92T + 23T^{2} \)
29 \( 1 + 10.7T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 8.15iT - 41T^{2} \)
43 \( 1 - 3.62T + 43T^{2} \)
47 \( 1 - 9.48iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 0.219iT - 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 8.72iT - 83T^{2} \)
89 \( 1 + 3.74iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.578270930418631420326558228490, −8.601840731277728256309379369763, −7.68442203481587007100637814638, −7.16505494867290926015047692850, −6.18875015918778189987741489985, −5.47643410663086022802143310620, −4.53262469587491989664500153455, −3.18148592295115947273218545871, −1.99355681980930722526384021400, −1.04807716952604171679047883287, 2.36707736700809061146951364622, 3.43724645364158732843358271231, 3.90081929874144743661926817262, 5.06023315259356229678288856367, 5.63487639634591663130656292659, 6.76661251358561272762080543881, 7.53645337125957682401121316142, 8.878691709664543722622019653435, 9.726191508574479279156623191906, 10.53635153518643962175830187208

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