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Properties

Label 2-980-28.27-c1-0-40
Degree $2$
Conductor $980$
Sign $0.802 - 0.596i$
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 − 0.153i)2-s + 3.02·3-s + (1.95 + 0.432i)4-s + i·5-s + (−4.25 − 0.465i)6-s + (−2.67 − 0.908i)8-s + 6.16·9-s + (0.153 − 1.40i)10-s + 1.19i·11-s + (5.91 + 1.30i)12-s + 4.83i·13-s + 3.02i·15-s + (3.62 + 1.68i)16-s − 2.54i·17-s + (−8.66 − 0.948i)18-s − 1.42·19-s + ⋯
L(s)  = 1  + (−0.994 − 0.108i)2-s + 1.74·3-s + (0.976 + 0.216i)4-s + 0.447i·5-s + (−1.73 − 0.190i)6-s + (−0.946 − 0.321i)8-s + 2.05·9-s + (0.0486 − 0.444i)10-s + 0.360i·11-s + (1.70 + 0.378i)12-s + 1.34i·13-s + 0.781i·15-s + (0.906 + 0.422i)16-s − 0.617i·17-s + (−2.04 − 0.223i)18-s − 0.326·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.802 - 0.596i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82499 + 0.603782i\)
\(L(\frac12)\) \(\approx\) \(1.82499 + 0.603782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.153i)T \)
5 \( 1 - iT \)
7 \( 1 \)
good3 \( 1 - 3.02T + 3T^{2} \)
11 \( 1 - 1.19iT - 11T^{2} \)
13 \( 1 - 4.83iT - 13T^{2} \)
17 \( 1 + 2.54iT - 17T^{2} \)
19 \( 1 + 1.42T + 19T^{2} \)
23 \( 1 - 5.80iT - 23T^{2} \)
29 \( 1 - 0.774T + 29T^{2} \)
31 \( 1 - 6.63T + 31T^{2} \)
37 \( 1 - 5.10T + 37T^{2} \)
41 \( 1 + 7.46iT - 41T^{2} \)
43 \( 1 - 1.38iT - 43T^{2} \)
47 \( 1 - 1.07T + 47T^{2} \)
53 \( 1 - 3.36T + 53T^{2} \)
59 \( 1 + 9.88T + 59T^{2} \)
61 \( 1 - 9.59iT - 61T^{2} \)
67 \( 1 + 10.5iT - 67T^{2} \)
71 \( 1 + 16.3iT - 71T^{2} \)
73 \( 1 + 0.107iT - 73T^{2} \)
79 \( 1 + 10.7iT - 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 3.94iT - 89T^{2} \)
97 \( 1 - 8.71iT - 97T^{2} \)
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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

−9.699452760896838609656581637379, −9.260260649004423046289981085675, −8.589574969358358576251086177275, −7.63099769066164382859594522617, −7.21440965908745223949938195387, −6.26845818759827740025699580613, −4.41945013240139207296101046984, −3.38321593957579651047895187745, −2.49249105085878610547722807393, −1.65139828917744795609734882896, 1.08073342314369534068772876422, 2.43879404050506736045606454806, 3.12883115208931453676482963645, 4.37113252473887083197331699205, 5.84404508221303768162116907714, 6.88050943925133463975912911089, 7.985287711924022059502659491518, 8.259584976736282643070291294499, 8.823089255960218651505348901794, 9.830331712348421751662704564157

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