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Properties

Label 2-980-7.2-c1-0-0
Degree $2$
Conductor $980$
Sign $0.605 - 0.795i$
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  + (−1.5 − 2.59i)3-s + (0.5 − 0.866i)5-s + (−3 + 5.19i)9-s + (2.5 + 4.33i)11-s − 3·13-s − 3·15-s + (0.5 + 0.866i)17-s + (−3 + 5.19i)19-s + (−3 + 5.19i)23-s + (−0.499 − 0.866i)25-s + 9·27-s − 9·29-s + (2 + 3.46i)31-s + (7.50 − 12.9i)33-s + (−1 + 1.73i)37-s + ⋯
L(s)  = 1  + (−0.866 − 1.49i)3-s + (0.223 − 0.387i)5-s + (−1 + 1.73i)9-s + (0.753 + 1.30i)11-s − 0.832·13-s − 0.774·15-s + (0.121 + 0.210i)17-s + (−0.688 + 1.19i)19-s + (−0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s + 1.73·27-s − 1.67·29-s + (0.359 + 0.622i)31-s + (1.30 − 2.26i)33-s + (−0.164 + 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526756 + 0.261127i\)
\(L(\frac12)\) \(\approx\) \(0.526756 + 0.261127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−10.10029021743968538023671421925, −9.374713437772091599844849458417, −8.140181850176752005969610980992, −7.44993758185122295633808126641, −6.79469758201643593411032712868, −5.91614061787608399566901158135, −5.19033671783751511397496297411, −3.99935389507306078874350177370, −2.09494057477336641200053054371, −1.48724060963417655401236893420, 0.30127134837684575171694354925, 2.64709451011125062993507827783, 3.81359052049937885899183199778, 4.52325463981775265974686942944, 5.58222151954653710203048705420, 6.14958816115782488649017942668, 7.13796872600580535725060506007, 8.563787333573754041189024056033, 9.267628747341843134584118262387, 9.934430994249639686033959697071

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