↑

Properties

Label 2-980-35.4-c1-0-6
Degree $2$
Conductor $980$
Sign $0.982 - 0.185i$
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  + (−2.23 + 0.133i)5-s + (−1.5 − 2.59i)9-s + 4i·13-s + (3.46 + 2i)17-s + (2 + 3.46i)19-s + (6.92 − 4i)23-s + (4.96 − 0.598i)25-s − 2·29-s + (4 − 6.92i)31-s + (6.92 − 4i)37-s + 6·41-s + 8i·43-s + (3.69 + 5.59i)45-s + (−6.92 + 4i)47-s + (−2 + 3.46i)59-s + ⋯
L(s)  = 1  + (−0.998 + 0.0599i)5-s + (−0.5 − 0.866i)9-s + 1.10i·13-s + (0.840 + 0.485i)17-s + (0.458 + 0.794i)19-s + (1.44 − 0.834i)23-s + (0.992 − 0.119i)25-s − 0.371·29-s + (0.718 − 1.24i)31-s + (1.13 − 0.657i)37-s + 0.937·41-s + 1.21i·43-s + (0.550 + 0.834i)45-s + (−1.01 + 0.583i)47-s + (−0.260 + 0.450i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.982 - 0.185i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.982 - 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26846 + 0.118757i\)
\(L(\frac12)\) \(\approx\) \(1.26846 + 0.118757i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.133i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.92 + 4i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + (6.92 - 4i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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

−9.881095382358240547002154553866, −9.199099342813155425546326455796, −8.310126054698841747483603129600, −7.58678863588756123840577605265, −6.63730428256926712623220216119, −5.84734254600871668037574197444, −4.53978395344829857762746738944, −3.77414100600725362662326140358, −2.78372263240156959442536848385, −0.962554427255944045818718986399, 0.845848652453346445710810900749, 2.78359223102205622521061339146, 3.47530894235330696219250094753, 4.96998853271478763571430683095, 5.29699790414913958053695680384, 6.77258980269466721507198350982, 7.64046123905680919581557778955, 8.128474708784298076304312557466, 9.050566794417287782413425083481, 10.03048080107623049725194204300

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