↑

Properties

Label 2-980-35.3-c1-0-17
Degree $2$
Conductor $980$
Sign $0.108 + 0.994i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.144 − 0.540i)3-s + (1.50 − 1.65i)5-s + (2.32 − 1.34i)9-s + (2.49 − 4.32i)11-s + (0.796 − 0.796i)13-s + (−1.11 − 0.575i)15-s + (−5.33 + 1.42i)17-s + (1.10 + 1.91i)19-s + (−1.89 + 7.08i)23-s + (−0.457 − 4.97i)25-s + (−2.25 − 2.25i)27-s − 2.10i·29-s + (−3.49 − 2.01i)31-s + (−2.70 − 0.723i)33-s + (9.23 + 2.47i)37-s + ⋯
L(s)  = 1  + (−0.0836 − 0.312i)3-s + (0.673 − 0.738i)5-s + (0.775 − 0.447i)9-s + (0.753 − 1.30i)11-s + (0.220 − 0.220i)13-s + (−0.286 − 0.148i)15-s + (−1.29 + 0.346i)17-s + (0.253 + 0.438i)19-s + (−0.395 + 1.47i)23-s + (−0.0914 − 0.995i)25-s + (−0.433 − 0.433i)27-s − 0.391i·29-s + (−0.627 − 0.362i)31-s + (−0.470 − 0.126i)33-s + (1.51 + 0.406i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.108 + 0.994i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.108 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35086 - 1.21086i\)
\(L(\frac12)\) \(\approx\) \(1.35086 - 1.21086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 + 1.65i)T \)
7 \( 1 \)
good3 \( 1 + (0.144 + 0.540i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.49 + 4.32i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.796 + 0.796i)T - 13iT^{2} \)
17 \( 1 + (5.33 - 1.42i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.89 - 7.08i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.10iT - 29T^{2} \)
31 \( 1 + (3.49 + 2.01i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.23 - 2.47i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 9.32iT - 41T^{2} \)
43 \( 1 + (-3.09 - 3.09i)T + 43iT^{2} \)
47 \( 1 + (-0.781 + 2.91i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.48 + 0.933i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.73 - 8.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.50 + 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.73 + 10.1i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.88T + 71T^{2} \)
73 \( 1 + (2.79 + 10.4i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.32 - 1.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.9 - 11.9i)T - 83iT^{2} \)
89 \( 1 + (1.81 + 3.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.67 - 8.67i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.581188153705297954050245870721, −9.092787816877817358699422543497, −8.246687014480377315124998274204, −7.25407790541464848162057670745, −6.10776589796053660413591660683, −5.82478901261521123115282950471, −4.41971225133709541760193442632, −3.58283882109084952619541388698, −1.95929183264896194414224136242, −0.905852862284589040585356639319, 1.72428702340613975687276322266, 2.65165947067439721655850196429, 4.21047990957067530061018128474, 4.70958088313362184885430137166, 6.05648808633705936115186659000, 6.88684865566464400658132210054, 7.35644410292126293269792294976, 8.764820793302869856270494372242, 9.543595209260014068372909336037, 10.10512525553457852578607546828

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