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Properties

Label 2-980-7.2-c1-0-5
Degree $2$
Conductor $980$
Sign $0.386 + 0.922i$
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 + (1 + 1.73i)11-s + 6·13-s + 3·15-s + (1 + 1.73i)17-s + (4.5 − 7.79i)23-s + (−0.499 − 0.866i)25-s + 9·27-s + 3·29-s + (1 + 1.73i)31-s + (3 − 5.19i)33-s + (−4 + 6.92i)37-s + (−9 − 15.5i)39-s + ⋯
L(s)  = 1  + (−0.866 − 1.49i)3-s + (−0.223 + 0.387i)5-s + (−1 + 1.73i)9-s + (0.301 + 0.522i)11-s + 1.66·13-s + 0.774·15-s + (0.242 + 0.420i)17-s + (0.938 − 1.62i)23-s + (−0.0999 − 0.173i)25-s + 1.73·27-s + 0.557·29-s + (0.179 + 0.311i)31-s + (0.522 − 0.904i)33-s + (−0.657 + 1.13i)37-s + (−1.44 − 2.49i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
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: \((2,\ 980,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.986053 - 0.655905i\)
\(L(\frac12)\) \(\approx\) \(0.986053 - 0.655905i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
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 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.12415045780528937101596296056, −8.575985217166682750136144618897, −8.199973389632290018009885314443, −6.89064595664227616231617602403, −6.69752257954642342614077714698, −5.83866256231900707754218541761, −4.74011430270501850512274903563, −3.35323353790531445569869501298, −1.96222052862298600103783583546, −0.869566947810801854820645664028, 0.998071215017101942980472863605, 3.35284352103996439862655159452, 3.90327264761187513524774144251, 4.95869957540646618009965506755, 5.66851783944335284476522793276, 6.40432213967838869900548633656, 7.74886919339459238350470747174, 9.019429307288767411934948838560, 9.140461885245802917817875130940, 10.31129931620928935996612885302

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