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Properties

Label 2-980-20.7-c1-0-22
Degree $2$
Conductor $980$
Sign $-0.820 + 0.572i$
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  + (0.0374 + 1.41i)2-s + (1.81 + 1.81i)3-s + (−1.99 + 0.105i)4-s + (0.00568 + 2.23i)5-s + (−2.49 + 2.63i)6-s + (−0.224 − 2.81i)8-s + 3.58i·9-s + (−3.16 + 0.0918i)10-s − 1.84i·11-s + (−3.81 − 3.43i)12-s + (−2.94 + 2.94i)13-s + (−4.04 + 4.06i)15-s + (3.97 − 0.423i)16-s + (2.17 + 2.17i)17-s + (−5.07 + 0.134i)18-s − 5.32·19-s + ⋯
L(s)  = 1  + (0.0264 + 0.999i)2-s + (1.04 + 1.04i)3-s + (−0.998 + 0.0529i)4-s + (0.00254 + 0.999i)5-s + (−1.01 + 1.07i)6-s + (−0.0793 − 0.996i)8-s + 1.19i·9-s + (−0.999 + 0.0290i)10-s − 0.555i·11-s + (−1.10 − 0.990i)12-s + (−0.817 + 0.817i)13-s + (−1.04 + 1.05i)15-s + (0.994 − 0.105i)16-s + (0.527 + 0.527i)17-s + (−1.19 + 0.0316i)18-s − 1.22·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.820 + 0.572i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.820 + 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496608 - 1.57990i\)
\(L(\frac12)\) \(\approx\) \(0.496608 - 1.57990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0374 - 1.41i)T \)
5 \( 1 + (-0.00568 - 2.23i)T \)
7 \( 1 \)
good3 \( 1 + (-1.81 - 1.81i)T + 3iT^{2} \)
11 \( 1 + 1.84iT - 11T^{2} \)
13 \( 1 + (2.94 - 2.94i)T - 13iT^{2} \)
17 \( 1 + (-2.17 - 2.17i)T + 17iT^{2} \)
19 \( 1 + 5.32T + 19T^{2} \)
23 \( 1 + (1.78 + 1.78i)T + 23iT^{2} \)
29 \( 1 - 6.35iT - 29T^{2} \)
31 \( 1 - 4.36iT - 31T^{2} \)
37 \( 1 + (2.83 + 2.83i)T + 37iT^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + (1.02 + 1.02i)T + 43iT^{2} \)
47 \( 1 + (-0.575 + 0.575i)T - 47iT^{2} \)
53 \( 1 + (-1.91 + 1.91i)T - 53iT^{2} \)
59 \( 1 - 4.22T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (-2.33 + 2.33i)T - 67iT^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (1.44 - 1.44i)T - 73iT^{2} \)
79 \( 1 - 7.91T + 79T^{2} \)
83 \( 1 + (0.227 + 0.227i)T + 83iT^{2} \)
89 \( 1 - 4.33iT - 89T^{2} \)
97 \( 1 + (-0.196 - 0.196i)T + 97iT^{2} \)
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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

−10.26719560565761311638203636183, −9.509173633578253771924101475935, −8.757059448296399674606023934878, −8.127456000976996508571427438732, −7.13418897905256960784185635297, −6.39683276684405933250015357582, −5.27759557744425971421281603736, −4.16271606537459724445544372331, −3.59154350019333902669988870313, −2.43953473967114103357766431439, 0.65980515141689291726300593345, 1.95150083537447275492871559011, 2.62828229851296085410554369720, 3.91500724524919642931833967974, 4.86253973006192713386032566173, 5.92638187272052374688738944652, 7.43254514559711203263500019244, 8.002047055798072080339912316859, 8.654757214720575452825255589366, 9.550427930653425930717539072373

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