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Properties

Label 2-980-35.4-c1-0-7
Degree $2$
Conductor $980$
Sign $0.0986 - 0.995i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)3-s + (1.63 + 1.52i)5-s + (−1.13 + 1.97i)11-s + 6.09i·13-s + (1.13 + 3.70i)15-s + (−4.13 − 2.38i)17-s + (2.13 + 3.70i)19-s + (0.774 − 0.447i)23-s + (0.362 + 4.98i)25-s − 5.19i·27-s + 3.27·29-s + (−2.13 + 3.70i)31-s + (−3.41 + 1.97i)33-s + (4.86 − 2.80i)37-s + (−5.27 + 9.13i)39-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (0.732 + 0.680i)5-s + (−0.342 + 0.594i)11-s + 1.68i·13-s + (0.293 + 0.955i)15-s + (−1.00 − 0.579i)17-s + (0.490 + 0.849i)19-s + (0.161 − 0.0932i)23-s + (0.0725 + 0.997i)25-s − 0.999i·27-s + 0.608·29-s + (−0.383 + 0.664i)31-s + (−0.594 + 0.342i)33-s + (0.799 − 0.461i)37-s + (−0.844 + 1.46i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0986 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0986 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.0986 - 0.995i$
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: \((2,\ 980,\ (\ :1/2),\ 0.0986 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65239 + 1.49675i\)
\(L(\frac12)\) \(\approx\) \(1.65239 + 1.49675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 1.52i)T \)
7 \( 1 \)
good3 \( 1 + (-1.5 - 0.866i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.13 - 1.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.09iT - 13T^{2} \)
17 \( 1 + (4.13 + 2.38i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.13 - 3.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.774 + 0.447i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 + (2.13 - 3.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.86 + 2.80i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + 6.50iT - 43T^{2} \)
47 \( 1 + (-1.86 + 1.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.41 + 3.70i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.13 - 3.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.774 - 1.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.0 + 6.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (-1.86 - 1.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.137 + 0.238i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.67iT - 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.982784201964126545534177290107, −9.292208936337802032592301731386, −8.916929361512379315220663251789, −7.64392029292046499230979032308, −6.83609233709056375197379581958, −6.04164448305357931036516488694, −4.75793072180921361689850446533, −3.87633730315401775075689056968, −2.74579446592009931401234472719, −1.94130971427855647793677836062, 0.945252816187647753011242282979, 2.38016032719902969222988574103, 3.06118209029767841100099964641, 4.58302768564089329542421197940, 5.53502414156249029738533042188, 6.29175966788611932923959960166, 7.62235158519014918214147507618, 8.137967816005220850276719423108, 8.889003701346648070957846850136, 9.582140836438989203666741664310

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