↑

Properties

Label 2-980-35.4-c1-0-16
Degree $2$
Conductor $980$
Sign $0.324 + 0.946i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.324 + 0.946i$
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: \((2,\ 980,\ (\ :1/2),\ 0.324 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34139 - 0.958388i\)
\(L(\frac12)\) \(\approx\) \(1.34139 - 0.958388i\)
\(L(\frac{3}{2})\) 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

−9.696152806773995496188564801886, −9.041052546255765224036858581061, −8.448071360066614889010003159673, −7.06778963287421324345818032641, −6.45697787158082993671958087882, −5.50618360276221769350501894068, −4.73781527171517681112547813408, −3.25511310276011531471851352211, −2.41481317178171931570428039069, −0.76086699125454369558400739701, 1.70214572125373996899278645018, 2.52183853143902469190831295643, 3.96993707716611406404570655210, 5.05341900623645401577263226458, 5.85801455012162503040410374454, 6.68566993326071901583519725304, 7.62185082612564095712188100269, 8.715649061867129605629859273195, 9.253517628863051982429364402022, 10.16417823794315622549665041763

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