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Properties

Label 2-980-7.4-c3-0-9
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  + (4 − 6.92i)3-s + (−2.5 − 4.33i)5-s + (−18.4 − 32.0i)9-s + (−14 + 24.2i)11-s − 82·13-s − 40·15-s + (−23 + 39.8i)17-s + (4 + 6.92i)19-s + (64 + 110. i)23-s + (−12.5 + 21.6i)25-s − 79.9·27-s + 174·29-s + (−76 + 131. i)31-s + (112 + 193. i)33-s + (145 + 251. i)37-s + ⋯
L(s)  = 1  + (0.769 − 1.33i)3-s + (−0.223 − 0.387i)5-s + (−0.685 − 1.18i)9-s + (−0.383 + 0.664i)11-s − 1.74·13-s − 0.688·15-s + (−0.328 + 0.568i)17-s + (0.0482 + 0.0836i)19-s + (0.580 + 1.00i)23-s + (−0.100 + 0.173i)25-s − 0.570·27-s + 1.11·29-s + (−0.440 + 0.762i)31-s + (0.590 + 1.02i)33-s + (0.644 + 1.11i)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

\(L(2)\) \(\approx\) \(1.142457525\)
\(L(\frac12)\) \(\approx\) \(1.142457525\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4 + 6.92i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (14 - 24.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 82T + 2.19e3T^{2} \)
17 \( 1 + (23 - 39.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-4 - 6.92i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-64 - 110. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 174T + 2.43e4T^{2} \)
31 \( 1 + (76 - 131. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-145 - 251. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 50T + 6.89e4T^{2} \)
43 \( 1 - 396T + 7.95e4T^{2} \)
47 \( 1 + (148 + 256. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-285 + 493. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (136 - 235. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (331 + 573. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (438 - 758. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 880T + 3.57e5T^{2} \)
73 \( 1 + (319 - 552. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-300 - 519. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 624T + 5.71e5T^{2} \)
89 \( 1 + (-349 - 604. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 754T + 9.12e5T^{2} \)
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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

−9.578088529916806497363549068851, −8.689698659518013643953939938711, −7.929665460074919838173975112306, −7.28756486163372603149852683016, −6.72002505763741299966201826139, −5.37797321273369175628809058382, −4.46429136085393199724295386055, −3.02239194052204178708633429548, −2.19956350640413109791498923297, −1.18867144414941949084938653413, 0.25887249713870444490313412509, 2.59544430257152017989158223305, 2.93792390374923940851245629061, 4.29958254139505781414555491345, 4.76413379407635841889615325983, 5.92500684544908649533181636006, 7.22309799664181432874700077140, 7.897868988688849070696951723080, 8.998130515166410650082154311448, 9.379234956705926604067740609834

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