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Properties

Label 2-980-28.27-c1-0-18
Degree $2$
Conductor $980$
Sign $0.475 - 0.879i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.475 - 0.879i$
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: \((2,\ 980,\ (\ :1/2),\ 0.475 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386082 + 0.230153i\)
\(L(\frac12)\) \(\approx\) \(0.386082 + 0.230153i\)
\(L(\frac{3}{2})\) 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

−10.28718837474594195019096775498, −9.499312936267187978274478790564, −8.592384535622678917319266606737, −7.21819002777721775289708802232, −6.88624741141006407409826161059, −6.03742206803704338360754631077, −5.27924874069704078107152294118, −4.04635796927074233221216909370, −2.30569928391142649297529787300, −0.853091938055869321148336069514, 0.52970971591702678161680242371, 1.66655205768209936100384408262, 3.48713887503132056797990642323, 4.91988416082240105923491440859, 5.70785623240316560587641706742, 6.36675297799163314957922500135, 7.43389568244093916382121492152, 8.017650538476340565726848960437, 9.314912054397237174605545344245, 9.979580087196258398308353148089

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