↑

Properties

Label 2-980-140.107-c1-0-18
Degree $2$
Conductor $980$
Sign $-0.382 - 0.924i$
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  + (−0.977 − 1.02i)2-s + (−0.346 + 1.29i)3-s + (−0.0888 + 1.99i)4-s + (1.92 − 1.13i)5-s + (1.65 − 0.909i)6-s + (2.12 − 1.86i)8-s + (1.04 + 0.605i)9-s + (−3.04 − 0.855i)10-s + (−4.33 + 2.50i)11-s + (−2.55 − 0.806i)12-s + (−3.27 + 3.27i)13-s + (0.802 + 2.88i)15-s + (−3.98 − 0.355i)16-s + (−0.873 + 3.25i)17-s + (−0.406 − 1.66i)18-s + (0.214 − 0.371i)19-s + ⋯
L(s)  = 1  + (−0.691 − 0.722i)2-s + (−0.199 + 0.745i)3-s + (−0.0444 + 0.999i)4-s + (0.861 − 0.508i)5-s + (0.677 − 0.371i)6-s + (0.752 − 0.658i)8-s + (0.349 + 0.201i)9-s + (−0.962 − 0.270i)10-s + (−1.30 + 0.754i)11-s + (−0.736 − 0.232i)12-s + (−0.908 + 0.908i)13-s + (0.207 + 0.743i)15-s + (−0.996 − 0.0887i)16-s + (−0.211 + 0.790i)17-s + (−0.0957 − 0.392i)18-s + (0.0491 − 0.0851i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.382 - 0.924i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.382 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360001 + 0.538555i\)
\(L(\frac12)\) \(\approx\) \(0.360001 + 0.538555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.977 + 1.02i)T \)
5 \( 1 + (-1.92 + 1.13i)T \)
7 \( 1 \)
good3 \( 1 + (0.346 - 1.29i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (4.33 - 2.50i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.27 - 3.27i)T - 13iT^{2} \)
17 \( 1 + (0.873 - 3.25i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.214 + 0.371i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.47 - 0.931i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.51iT - 29T^{2} \)
31 \( 1 + (0.353 - 0.203i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.33 - 1.16i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.67T + 41T^{2} \)
43 \( 1 + (4.01 + 4.01i)T + 43iT^{2} \)
47 \( 1 + (-2.60 - 9.74i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.10 - 0.296i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.93 - 3.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.11 - 10.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.3 + 3.03i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.2iT - 71T^{2} \)
73 \( 1 + (5.17 + 1.38i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.0987 + 0.171i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.86 + 1.86i)T + 83iT^{2} \)
89 \( 1 + (-6.11 - 3.52i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.24 + 5.24i)T + 97iT^{2} \)
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

−10.17990704824287686019526051376, −9.675163733149239924978824176861, −8.955650454164591086107609450534, −7.903893255597771659471519672822, −7.16053333898714722876077212580, −5.81159299617670129439764104307, −4.69318842452883038961168064273, −4.22019630156507925985326002018, −2.53661583987058556014638709004, −1.77029164490799982217354865953, 0.36134972982272707366935924676, 1.88218677414507667079037440762, 2.96094102274974801282066261109, 5.00322601057749533607576349580, 5.62942902076226809021364486852, 6.47158933757956989163347902793, 7.29145880946703747190787432466, 7.79930010620495554118540700900, 8.837439508269826176470153593153, 9.803673795469768490787268136212

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