↑

Properties

Label 2-980-140.107-c1-0-13
Degree $2$
Conductor $980$
Sign $-0.432 + 0.901i$
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  + (1.04 + 0.952i)2-s + (−0.727 + 2.71i)3-s + (0.185 + 1.99i)4-s + (−0.178 − 2.22i)5-s + (−3.34 + 2.14i)6-s + (−1.70 + 2.25i)8-s + (−4.24 − 2.44i)9-s + (1.93 − 2.50i)10-s + (−2.75 + 1.59i)11-s + (−5.54 − 0.944i)12-s + (−2.41 + 2.41i)13-s + (6.18 + 1.13i)15-s + (−3.93 + 0.739i)16-s + (0.600 − 2.24i)17-s + (−2.10 − 6.60i)18-s + (1.39 − 2.42i)19-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)2-s + (−0.419 + 1.56i)3-s + (0.0928 + 0.995i)4-s + (−0.0799 − 0.996i)5-s + (−1.36 + 0.875i)6-s + (−0.601 + 0.798i)8-s + (−1.41 − 0.816i)9-s + (0.612 − 0.790i)10-s + (−0.831 + 0.479i)11-s + (−1.59 − 0.272i)12-s + (−0.670 + 0.670i)13-s + (1.59 + 0.293i)15-s + (−0.982 + 0.184i)16-s + (0.145 − 0.543i)17-s + (−0.495 − 1.55i)18-s + (0.320 − 0.555i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.432 + 0.901i$
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: \((2,\ 980,\ (\ :1/2),\ -0.432 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.472112 - 0.750134i\)
\(L(\frac12)\) \(\approx\) \(0.472112 - 0.750134i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 - 0.952i)T \)
5 \( 1 + (0.178 + 2.22i)T \)
7 \( 1 \)
good3 \( 1 + (0.727 - 2.71i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (2.75 - 1.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.41 - 2.41i)T - 13iT^{2} \)
17 \( 1 + (-0.600 + 2.24i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.39 + 2.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.19 - 1.39i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 1.72iT - 29T^{2} \)
31 \( 1 + (3.01 - 1.74i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.32 + 0.623i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.72T + 41T^{2} \)
43 \( 1 + (-3.96 - 3.96i)T + 43iT^{2} \)
47 \( 1 + (-1.63 - 6.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-12.6 - 3.37i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.951 - 1.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.83 - 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.67 - 1.51i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.562iT - 71T^{2} \)
73 \( 1 + (-3.23 - 0.866i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.13 - 7.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.38 + 4.38i)T + 83iT^{2} \)
89 \( 1 + (2.51 + 1.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 10.9i)T + 97iT^{2} \)
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

−10.48714314641261628588096521847, −9.551193346208430608639420020762, −9.096010518100545902252986314360, −8.026382863691538034589621637582, −7.17949880974021835105320587078, −5.79536846620375085634154268006, −5.22667418615504490160113873019, −4.52105588120519335656575166839, −3.97693033348664322493430523516, −2.59632286120666058924449036160, 0.31089637524164340171861151196, 1.91166964920781353868994530964, 2.68880072600973465034191501211, 3.75088574033981894312899518210, 5.40024953219061197110550125456, 5.90096451138307589487249453781, 6.76716645786922922505193298553, 7.54421135417336133121845373230, 8.274498964372337568883364421278, 9.881302736433286847242820108417

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