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Properties

Label 2-980-35.12-c1-0-18
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: $\chi_{980} (117, \cdot )$
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

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

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