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Properties

Label 2-980-7.4-c3-0-17
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  + (−2.5 + 4.33i)3-s + (−2.5 − 4.33i)5-s + (0.999 + 1.73i)9-s + (−7.5 + 12.9i)11-s − 17·13-s + 25.0·15-s + (61.5 − 106. i)17-s + (43 + 74.4i)19-s + (−27 − 46.7i)23-s + (−12.5 + 21.6i)25-s − 144.·27-s − 177·29-s + (106 − 183. i)31-s + (−37.5 − 64.9i)33-s + (−37 − 64.0i)37-s + ⋯
L(s)  = 1  + (−0.481 + 0.833i)3-s + (−0.223 − 0.387i)5-s + (0.0370 + 0.0641i)9-s + (−0.205 + 0.356i)11-s − 0.362·13-s + 0.430·15-s + (0.877 − 1.51i)17-s + (0.519 + 0.899i)19-s + (−0.244 − 0.423i)23-s + (−0.100 + 0.173i)25-s − 1.03·27-s − 1.13·29-s + (0.614 − 1.06i)31-s + (−0.197 − 0.342i)33-s + (−0.164 − 0.284i)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: $\chi_{980} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :3/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.482313326\)
\(L(\frac12)\) \(\approx\) \(1.482313326\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 4.33i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (7.5 - 12.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 17T + 2.19e3T^{2} \)
17 \( 1 + (-61.5 + 106. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-43 - 74.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (27 + 46.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 177T + 2.43e4T^{2} \)
31 \( 1 + (-106 + 183. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (37 + 64.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 444T + 6.89e4T^{2} \)
43 \( 1 + 46T + 7.95e4T^{2} \)
47 \( 1 + (-235.5 - 407. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-90 + 155. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-72 + 124. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (188 + 325. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (178 - 308. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 48T + 3.57e5T^{2} \)
73 \( 1 + (-409 + 708. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (44.5 + 77.0i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 780T + 5.71e5T^{2} \)
89 \( 1 + (-570 - 987. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 169T + 9.12e5T^{2} \)
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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

−9.653878959646510431425686621953, −9.288354624026992367087186359882, −7.76464771848965446265953143492, −7.53961193186302476129887403690, −6.02649579317656860332434438345, −5.23654795994774671049217466886, −4.56098364758905647478419462778, −3.62623176255686470454202195056, −2.29217034027863760794250280753, −0.70962602446395108966253659547, 0.63762030867160650430192123642, 1.74953374543297199890480009619, 3.09004760860185324833719476248, 4.08042218372562313237119552194, 5.46265737014274616727218366797, 6.09576320664532114113630058994, 7.06879059015736121877036787552, 7.60128254261588492720532025644, 8.540656192203946908402475461800, 9.573492661330008689593635436760

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