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Properties

Label 2-980-7.2-c3-0-23
Degree $2$
Conductor $980$
Sign $0.701 + 0.712i$
Analytic cond. $57.8218$
Root an. cond. $7.60406$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−2.5 + 4.33i)5-s + (13 − 22.5i)9-s + (3.5 + 6.06i)11-s + 23·13-s − 5·15-s + (−12.5 − 21.6i)17-s + (−31 + 53.6i)19-s + (43 − 74.4i)23-s + (−12.5 − 21.6i)25-s + 53·27-s − 29·29-s + (−6 − 10.3i)31-s + (−3.5 + 6.06i)33-s + (75 − 129. i)37-s + ⋯
L(s)  = 1  + (0.0962 + 0.166i)3-s + (−0.223 + 0.387i)5-s + (0.481 − 0.833i)9-s + (0.0959 + 0.166i)11-s + 0.490·13-s − 0.0860·15-s + (−0.178 − 0.308i)17-s + (−0.374 + 0.648i)19-s + (0.389 − 0.675i)23-s + (−0.100 − 0.173i)25-s + 0.377·27-s − 0.185·29-s + (−0.0347 − 0.0602i)31-s + (−0.0184 + 0.0319i)33-s + (0.333 − 0.577i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(57.8218\)
Root analytic conductor: \(7.60406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :3/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.886319868\)
\(L(\frac12)\) \(\approx\) \(1.886319868\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-3.5 - 6.06i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 23T + 2.19e3T^{2} \)
17 \( 1 + (12.5 + 21.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (31 - 53.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-43 + 74.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 29T + 2.43e4T^{2} \)
31 \( 1 + (6 + 10.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-75 + 129. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 204T + 6.89e4T^{2} \)
43 \( 1 + 178T + 7.95e4T^{2} \)
47 \( 1 + (-16.5 + 28.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (226 + 391. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-60 - 103. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-460 + 796. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-150 - 259. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 520T + 3.57e5T^{2} \)
73 \( 1 + (-185 - 320. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-506.5 + 877. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 636T + 5.71e5T^{2} \)
89 \( 1 + (-146 + 252. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.38e3T + 9.12e5T^{2} \)
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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

−9.557362254803801152961214019553, −8.720610361147025841921867721876, −7.87517708968028144757327102986, −6.82098372492757166277130631601, −6.32519153791153022071215226233, −5.07058406709113345321093074028, −4.01007950189869516411631181433, −3.30222370538920811756429902841, −1.95076871011351465883411178091, −0.54768538601242965769033525709, 1.03431446025681599126858838319, 2.13292438950492262601301988211, 3.43499261969073760872123636792, 4.48882180682028855818488605379, 5.27421459406670266095904731332, 6.38788337774839780455465706008, 7.26682636281730129561541847335, 8.109516767940463180395512845363, 8.776158133476374440988373415558, 9.680829146742512283245697314442

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