↑

Properties

Label 2-980-7.2-c1-0-7
Degree $2$
Conductor $980$
Sign $0.947 + 0.318i$
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  + (0.207 + 0.358i)3-s + (0.5 − 0.866i)5-s + (1.41 − 2.44i)9-s + (1.91 + 3.31i)11-s + 3.58·13-s + 0.414·15-s + (−3.20 − 5.55i)17-s + (−1.82 + 3.16i)19-s + (−0.292 + 0.507i)23-s + (−0.499 − 0.866i)25-s + 2.41·27-s + 6.65·29-s + (−2.29 − 3.97i)31-s + (−0.792 + 1.37i)33-s + (1.70 − 2.95i)37-s + ⋯
L(s)  = 1  + (0.119 + 0.207i)3-s + (0.223 − 0.387i)5-s + (0.471 − 0.816i)9-s + (0.577 + 0.999i)11-s + 0.994·13-s + 0.106·15-s + (−0.777 − 1.34i)17-s + (−0.419 + 0.726i)19-s + (−0.0610 + 0.105i)23-s + (−0.0999 − 0.173i)25-s + 0.464·27-s + 1.23·29-s + (−0.411 − 0.713i)31-s + (−0.138 + 0.239i)33-s + (0.280 − 0.486i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.947 + 0.318i$
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: \((2,\ 980,\ (\ :1/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85902 - 0.304035i\)
\(L(\frac12)\) \(\approx\) \(1.85902 - 0.304035i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.207 - 0.358i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.91 - 3.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.58T + 13T^{2} \)
17 \( 1 + (3.20 + 5.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.82 - 3.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.292 - 0.507i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.65T + 29T^{2} \)
31 \( 1 + (2.29 + 3.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.70 + 2.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.585T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + (-4.44 + 7.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.87 - 3.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.70 + 2.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.58 - 4.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.53 - 9.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.48T + 71T^{2} \)
73 \( 1 + (2.58 + 4.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.57 - 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (8.48 - 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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

−9.755564501773977877861143233340, −9.229629267730791441607011150082, −8.512474783474903063040389069562, −7.31384207830101936810137953844, −6.60486821115730914193114835539, −5.68333913634415324527727453250, −4.43548084738216714837978559823, −3.90886430637287427107357005214, −2.41997427598441616360141008052, −1.05858090621102130325001002481, 1.31757867481311323134413049381, 2.55069355491189466451625478203, 3.74586305369303139155045424295, 4.68056568379224875526112334212, 6.05530963843593604773144146971, 6.46001693798905485993066490741, 7.55229397701890474673389881837, 8.530127118067732750843711582064, 8.944427066210856015270695129305, 10.27802181131232167316296589124

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