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Properties

Label 2-980-140.67-c1-0-59
Degree $2$
Conductor $980$
Sign $0.728 + 0.684i$
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  + (−1.24 − 0.675i)2-s + (0.402 − 0.107i)3-s + (1.08 + 1.67i)4-s + (2.22 − 0.188i)5-s + (−0.572 − 0.137i)6-s + (−0.219 − 2.81i)8-s + (−2.44 + 1.41i)9-s + (−2.89 − 1.26i)10-s + (0.725 + 0.418i)11-s + (0.618 + 0.557i)12-s + (1.16 − 1.16i)13-s + (0.875 − 0.316i)15-s + (−1.63 + 3.65i)16-s + (4.94 − 1.32i)17-s + (3.99 − 0.103i)18-s + (−2.91 − 5.05i)19-s + ⋯
L(s)  = 1  + (−0.878 − 0.477i)2-s + (0.232 − 0.0622i)3-s + (0.544 + 0.838i)4-s + (0.996 − 0.0843i)5-s + (−0.233 − 0.0561i)6-s + (−0.0777 − 0.996i)8-s + (−0.815 + 0.471i)9-s + (−0.915 − 0.401i)10-s + (0.218 + 0.126i)11-s + (0.178 + 0.160i)12-s + (0.322 − 0.322i)13-s + (0.226 − 0.0815i)15-s + (−0.407 + 0.913i)16-s + (1.20 − 0.321i)17-s + (0.941 − 0.0244i)18-s + (−0.669 − 1.15i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.728 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26331 - 0.500278i\)
\(L(\frac12)\) \(\approx\) \(1.26331 - 0.500278i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.675i)T \)
5 \( 1 + (-2.22 + 0.188i)T \)
7 \( 1 \)
good3 \( 1 + (-0.402 + 0.107i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.725 - 0.418i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.16 + 1.16i)T - 13iT^{2} \)
17 \( 1 + (-4.94 + 1.32i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.91 + 5.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.896 + 3.34i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.00iT - 29T^{2} \)
31 \( 1 + (-7.03 - 4.06i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.190 - 0.711i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.0958T + 41T^{2} \)
43 \( 1 + (-4.87 - 4.87i)T + 43iT^{2} \)
47 \( 1 + (-5.28 - 1.41i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.43 - 12.8i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.46 + 7.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.919 + 1.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.138 - 0.515i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (1.55 + 5.78i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.30 + 9.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.36 - 4.36i)T + 83iT^{2} \)
89 \( 1 + (-2.50 + 1.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.24 + 4.24i)T + 97iT^{2} \)
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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

−9.905302909073939152916390170561, −9.045426286934995727711916624367, −8.493960957897808052336299061553, −7.62439043032364566941844096248, −6.58315565811558936575313314880, −5.76100656784571279464689615986, −4.54544932151022537173697098580, −2.99148479490649970768712663072, −2.40109661656396986685430736607, −0.985035810505500395075113037608, 1.20838594562043903699333526767, 2.39245072327220017757726656545, 3.65029825115440889132389651936, 5.35859998873283586205036585598, 5.94627886834842566771709825320, 6.63191677792158216767919797208, 7.75231843569340769469231997173, 8.590628378314308430852817639796, 9.166617018400989547425480877934, 10.00290421344671710115295301620

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