↑

Properties

Label 2-980-20.3-c1-0-21
Degree $2$
Conductor $980$
Sign $-0.996 + 0.0781i$
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  + (0.566 + 1.29i)2-s + (0.792 − 0.792i)3-s + (−1.35 + 1.46i)4-s + (−0.427 + 2.19i)5-s + (1.47 + 0.578i)6-s + (−2.67 − 0.928i)8-s + 1.74i·9-s + (−3.08 + 0.689i)10-s + 2.98i·11-s + (0.0868 + 2.24i)12-s + (−4.05 − 4.05i)13-s + (1.40 + 2.07i)15-s + (−0.309 − 3.98i)16-s + (−1.68 + 1.68i)17-s + (−2.25 + 0.987i)18-s + 2.51·19-s + ⋯
L(s)  = 1  + (0.400 + 0.916i)2-s + (0.457 − 0.457i)3-s + (−0.679 + 0.733i)4-s + (−0.191 + 0.981i)5-s + (0.602 + 0.236i)6-s + (−0.944 − 0.328i)8-s + 0.581i·9-s + (−0.975 + 0.218i)10-s + 0.899i·11-s + (0.0250 + 0.646i)12-s + (−1.12 − 1.12i)13-s + (0.361 + 0.536i)15-s + (−0.0774 − 0.996i)16-s + (−0.409 + 0.409i)17-s + (−0.532 + 0.232i)18-s + 0.576·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.996 + 0.0781i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.996 + 0.0781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0507461 - 1.29593i\)
\(L(\frac12)\) \(\approx\) \(0.0507461 - 1.29593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.566 - 1.29i)T \)
5 \( 1 + (0.427 - 2.19i)T \)
7 \( 1 \)
good3 \( 1 + (-0.792 + 0.792i)T - 3iT^{2} \)
11 \( 1 - 2.98iT - 11T^{2} \)
13 \( 1 + (4.05 + 4.05i)T + 13iT^{2} \)
17 \( 1 + (1.68 - 1.68i)T - 17iT^{2} \)
19 \( 1 - 2.51T + 19T^{2} \)
23 \( 1 + (4.24 - 4.24i)T - 23iT^{2} \)
29 \( 1 + 2.55iT - 29T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + (3.37 - 3.37i)T - 37iT^{2} \)
41 \( 1 - 7.93T + 41T^{2} \)
43 \( 1 + (7.62 - 7.62i)T - 43iT^{2} \)
47 \( 1 + (-2.09 - 2.09i)T + 47iT^{2} \)
53 \( 1 + (-1.80 - 1.80i)T + 53iT^{2} \)
59 \( 1 + 2.09T + 59T^{2} \)
61 \( 1 - 1.90T + 61T^{2} \)
67 \( 1 + (-2.19 - 2.19i)T + 67iT^{2} \)
71 \( 1 - 8.09iT - 71T^{2} \)
73 \( 1 + (6.89 + 6.89i)T + 73iT^{2} \)
79 \( 1 - 8.06T + 79T^{2} \)
83 \( 1 + (5.99 - 5.99i)T - 83iT^{2} \)
89 \( 1 - 2.05iT - 89T^{2} \)
97 \( 1 + (6.63 - 6.63i)T - 97iT^{2} \)
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

−10.14882507451474171544963660576, −9.645798249517228397041450142339, −8.302960617383561221460695186816, −7.57311416914847572379063103689, −7.37375935464770803031489583445, −6.29105404216472065627567736853, −5.33606274647357604086243932040, −4.34674760103215815846867199700, −3.15893025118583444692328531810, −2.27425719278006456878514297540, 0.47345582125362518125314857265, 2.02150358921523590130496215833, 3.24614625655133590079894831973, 4.15626929912051175038157092204, 4.82955919511337001483418454943, 5.80190045526627979957936189303, 6.97447759668994268224265150574, 8.407334515206537201719180557018, 8.965500630050285345290126184829, 9.529151177475322206247600675704

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