Properties

Label 2-980-980.239-c0-0-0
Degree $2$
Conductor $980$
Sign $0.518 + 0.855i$
Analytic cond. $0.489083$
Root an. cond. $0.699345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 + 0.781i)2-s + (−1.62 − 0.781i)3-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (1.62 − 0.781i)6-s + (0.222 + 0.974i)7-s + (0.900 + 0.433i)8-s + (1.40 + 1.75i)9-s + (0.900 − 0.433i)10-s + (−0.400 + 1.75i)12-s + (−0.900 − 0.433i)14-s + (1.12 + 1.40i)15-s + (−0.900 + 0.433i)16-s − 2.24·18-s + (−0.222 + 0.974i)20-s + (0.400 − 1.75i)21-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−1.62 − 0.781i)3-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (1.62 − 0.781i)6-s + (0.222 + 0.974i)7-s + (0.900 + 0.433i)8-s + (1.40 + 1.75i)9-s + (0.900 − 0.433i)10-s + (−0.400 + 1.75i)12-s + (−0.900 − 0.433i)14-s + (1.12 + 1.40i)15-s + (−0.900 + 0.433i)16-s − 2.24·18-s + (−0.222 + 0.974i)20-s + (0.400 − 1.75i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.518 + 0.855i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.518 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2808149896\)
\(L(\frac12)\) \(\approx\) \(0.2808149896\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-0.222 - 0.974i)T \)
good3 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
29 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.900 + 0.433i)T^{2} \)
41 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
43 \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.222 - 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.24063302857148971333138762387, −8.934767566004377191802902626094, −8.268402824316746930758720506158, −7.44191489209069009029912107714, −6.66424095516793258767757267778, −5.89869047706462077277749229757, −5.16251400027676591725366808021, −4.41152545774057166764440618238, −1.99739946321435648735086389024, −0.50068067144684696359766649551, 1.12227874522549266783737319679, 3.32776932722988761998532540699, 4.12075175324556659238615922478, 4.76895444294960565116112356973, 6.05615879241325725409381191693, 7.20004402688021694384221563091, 7.63952502730630920954170155093, 8.969351617873892448132998839090, 9.980612453167528991655488616843, 10.43919929179297851213173250094

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