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Properties

Label 2-980-7.2-c3-0-27
Degree $2$
Conductor $980$
Sign $0.386 + 0.922i$
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  + (0.154 + 0.268i)3-s + (2.5 − 4.33i)5-s + (13.4 − 23.2i)9-s + (−8.30 − 14.3i)11-s + 91.6·13-s + 1.54·15-s + (13.8 + 23.9i)17-s + (39.3 − 68.0i)19-s + (−18.4 + 31.9i)23-s + (−12.5 − 21.6i)25-s + 16.6·27-s − 209.·29-s + (89.1 + 154. i)31-s + (2.57 − 4.45i)33-s + (−68.5 + 118. i)37-s + ⋯
L(s)  = 1  + (0.0297 + 0.0515i)3-s + (0.223 − 0.387i)5-s + (0.498 − 0.862i)9-s + (−0.227 − 0.394i)11-s + 1.95·13-s + 0.0266·15-s + (0.197 + 0.341i)17-s + (0.474 − 0.822i)19-s + (−0.167 + 0.289i)23-s + (−0.100 − 0.173i)25-s + 0.118·27-s − 1.34·29-s + (0.516 + 0.894i)31-s + (0.0135 − 0.0235i)33-s + (−0.304 + 0.527i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
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: \((2,\ 980,\ (\ :3/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.442933839\)
\(L(\frac12)\) \(\approx\) \(2.442933839\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.154 - 0.268i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (8.30 + 14.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 91.6T + 2.19e3T^{2} \)
17 \( 1 + (-13.8 - 23.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-39.3 + 68.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (18.4 - 31.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 209.T + 2.43e4T^{2} \)
31 \( 1 + (-89.1 - 154. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (68.5 - 118. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 225.T + 6.89e4T^{2} \)
43 \( 1 - 502.T + 7.95e4T^{2} \)
47 \( 1 + (-225. + 391. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (253. + 439. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (343. + 594. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (275. - 477. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-90.7 - 157. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 832.T + 3.57e5T^{2} \)
73 \( 1 + (170. + 295. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (89.6 - 155. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 708.T + 5.71e5T^{2} \)
89 \( 1 + (-434. + 752. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 801.T + 9.12e5T^{2} \)
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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

−9.276108729369820071337414448162, −8.796736419889250176704324987958, −7.907646418123903917868984707770, −6.77810668119625650544203162806, −6.05889544981514061432803417803, −5.18531515112882828008638995885, −3.93161908692107911445844035055, −3.28781598140749693175184521609, −1.63443757683588677024058812551, −0.69530804095716127397180001933, 1.19820880317996113694215721389, 2.22934373085073754533585797002, 3.49640639115996367059006491458, 4.39698263629386401609588472359, 5.62829415329962219075192681478, 6.23158644586393399413708606333, 7.44985732723032545058528841758, 7.905967780029294157007219163626, 9.006578756178002595117102570908, 9.795184946351451958266276756671

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