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Properties

Label 2-980-140.107-c1-0-17
Degree $2$
Conductor $980$
Sign $-0.0927 - 0.995i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0927 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0927 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.0927 - 0.995i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.0927 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.426238 + 0.467782i\)
\(L(\frac12)\) \(\approx\) \(0.426238 + 0.467782i\)
\(L(\frac{3}{2})\) 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

−10.15757989777119907504123499972, −9.513521425771650239361183117563, −8.370867080570982824935537297724, −7.66959564625185799036164597174, −6.97855305654211242891617513842, −6.52871461825659940946780722736, −5.26486965898974366454296309465, −3.50482964484870912999735107196, −2.24215524077296515696760163965, −1.83046439404696960479768163941, 0.35385841330494118269166732774, 2.08699904831721410309118090488, 3.25214697419948645501654888857, 4.58985874413780110220546812539, 5.44017359800563743315900033512, 6.30317109772652818870855588811, 7.79200304375774897336309158452, 8.116545003216602299065023310971, 9.198191330268540120655315010292, 9.708402851383755061350104002721

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