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Properties

Label 2-980-140.123-c1-0-70
Degree $2$
Conductor $980$
Sign $-0.998 + 0.0504i$
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  + (−1.40 + 0.144i)2-s + (−0.466 − 1.74i)3-s + (1.95 − 0.406i)4-s + (−0.858 + 2.06i)5-s + (0.907 + 2.38i)6-s + (−2.69 + 0.854i)8-s + (−0.213 + 0.122i)9-s + (0.910 − 3.02i)10-s + (−2.91 − 1.68i)11-s + (−1.62 − 3.21i)12-s + (2.69 + 2.69i)13-s + (3.99 + 0.531i)15-s + (3.66 − 1.59i)16-s + (0.587 + 2.19i)17-s + (0.281 − 0.203i)18-s + (−0.601 − 1.04i)19-s + ⋯
L(s)  = 1  + (−0.994 + 0.102i)2-s + (−0.269 − 1.00i)3-s + (0.979 − 0.203i)4-s + (−0.384 + 0.923i)5-s + (0.370 + 0.972i)6-s + (−0.953 + 0.302i)8-s + (−0.0710 + 0.0409i)9-s + (0.287 − 0.957i)10-s + (−0.878 − 0.507i)11-s + (−0.467 − 0.929i)12-s + (0.746 + 0.746i)13-s + (1.03 + 0.137i)15-s + (0.917 − 0.397i)16-s + (0.142 + 0.531i)17-s + (0.0664 − 0.0480i)18-s + (−0.138 − 0.239i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.998 + 0.0504i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.998 + 0.0504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00473074 - 0.187397i\)
\(L(\frac12)\) \(\approx\) \(0.00473074 - 0.187397i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.144i)T \)
5 \( 1 + (0.858 - 2.06i)T \)
7 \( 1 \)
good3 \( 1 + (0.466 + 1.74i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.91 + 1.68i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.69 - 2.69i)T + 13iT^{2} \)
17 \( 1 + (-0.587 - 2.19i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.601 + 1.04i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.37 + 1.97i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 6.53iT - 29T^{2} \)
31 \( 1 + (-3.65 - 2.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.72 + 0.729i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.68T + 41T^{2} \)
43 \( 1 + (1.90 - 1.90i)T - 43iT^{2} \)
47 \( 1 + (-2.77 + 10.3i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (12.6 - 3.38i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.60 - 6.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.61 + 4.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.338 - 0.0907i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + (13.9 - 3.74i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.05 - 7.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.4 - 11.4i)T - 83iT^{2} \)
89 \( 1 + (6.54 - 3.77i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0167 - 0.0167i)T - 97iT^{2} \)
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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

−9.700547957751395733164040218938, −8.390452791819684211878165792929, −7.969228814744705797860606614894, −7.13443662615558209259843765210, −6.34882580061184159606475627954, −5.92025178018079432232139981292, −4.01115115518874752459889619973, −2.71526014116417401188634567276, −1.69643325228264302451459988028, −0.12271201769942989679116029461, 1.54497109971324666020961275571, 3.14296134130816404795066413303, 4.22586027262484855683707693729, 5.20172856303263004538967911511, 6.02575654599507387551208826466, 7.48994301455520896279678335199, 7.998055198338730725691161354098, 8.869377072527852852919793471915, 9.687200255827739475792120524821, 10.25264199442523854033625413305

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