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Properties

Label 2-980-35.3-c1-0-6
Degree $2$
Conductor $980$
Sign $-0.771 - 0.636i$
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  + (0.769 + 2.87i)3-s + (2.17 + 0.515i)5-s + (−5.05 + 2.92i)9-s + (−1.48 + 2.57i)11-s + (2.81 − 2.81i)13-s + (0.194 + 6.64i)15-s + (−7.55 + 2.02i)17-s + (−0.608 − 1.05i)19-s + (−1.90 + 7.10i)23-s + (4.46 + 2.24i)25-s + (−5.97 − 5.97i)27-s − 3.02i·29-s + (7.07 + 4.08i)31-s + (−8.54 − 2.28i)33-s + (2.42 + 0.650i)37-s + ⋯
L(s)  = 1  + (0.444 + 1.65i)3-s + (0.973 + 0.230i)5-s + (−1.68 + 0.973i)9-s + (−0.448 + 0.776i)11-s + (0.779 − 0.779i)13-s + (0.0503 + 1.71i)15-s + (−1.83 + 0.490i)17-s + (−0.139 − 0.241i)19-s + (−0.396 + 1.48i)23-s + (0.893 + 0.448i)25-s + (−1.14 − 1.14i)27-s − 0.562i·29-s + (1.27 + 0.733i)31-s + (−1.48 − 0.398i)33-s + (0.399 + 0.106i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.771 - 0.636i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646933 + 1.79952i\)
\(L(\frac12)\) \(\approx\) \(0.646933 + 1.79952i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 0.515i)T \)
7 \( 1 \)
good3 \( 1 + (-0.769 - 2.87i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.48 - 2.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.81 + 2.81i)T - 13iT^{2} \)
17 \( 1 + (7.55 - 2.02i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.608 + 1.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.90 - 7.10i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.02iT - 29T^{2} \)
31 \( 1 + (-7.07 - 4.08i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.42 - 0.650i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.93iT - 41T^{2} \)
43 \( 1 + (2.28 + 2.28i)T + 43iT^{2} \)
47 \( 1 + (-0.982 + 3.66i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.77 + 0.742i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.18 + 5.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.4 + 6.01i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.16 - 11.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.97T + 71T^{2} \)
73 \( 1 + (2.00 + 7.48i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.77 + 1.02i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.04 - 2.04i)T - 83iT^{2} \)
89 \( 1 + (8.96 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.07 + 4.07i)T + 97iT^{2} \)
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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

−10.07410808994396151498059095463, −9.766573226369688835708504097151, −8.810949800098518259320355021602, −8.221020970055758602983880362829, −6.81049514679585935340375871814, −5.80686862249836066313435748017, −4.98542350298215055888596732485, −4.15891036468382392056398437122, −3.10916555492910608965744142449, −2.10427664497818662934301537770, 0.819131633835902870025820469356, 2.09918446242474458703693125849, 2.67636254954314215005124724985, 4.34340890929650027574683250125, 5.71365407920603130940093135207, 6.51050146833611383170958041273, 6.84856393579923604042624749058, 8.241523934355227770634613435119, 8.583185897120206267762218779763, 9.372164591216472497762374712926

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