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Properties

Label 2-980-980.499-c0-0-1
Degree $2$
Conductor $980$
Sign $0.117 - 0.993i$
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  + (0.733 + 0.680i)2-s + (0.722 + 0.108i)3-s + (0.0747 + 0.997i)4-s + (0.365 + 0.930i)5-s + (0.455 + 0.571i)6-s + (−0.0747 − 0.997i)7-s + (−0.623 + 0.781i)8-s + (−0.445 − 0.137i)9-s + (−0.365 + 0.930i)10-s + (−0.0546 + 0.728i)12-s + (0.623 − 0.781i)14-s + (0.162 + 0.712i)15-s + (−0.988 + 0.149i)16-s + (−0.233 − 0.403i)18-s + (−0.900 + 0.433i)20-s + (0.0546 − 0.728i)21-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (0.722 + 0.108i)3-s + (0.0747 + 0.997i)4-s + (0.365 + 0.930i)5-s + (0.455 + 0.571i)6-s + (−0.0747 − 0.997i)7-s + (−0.623 + 0.781i)8-s + (−0.445 − 0.137i)9-s + (−0.365 + 0.930i)10-s + (−0.0546 + 0.728i)12-s + (0.623 − 0.781i)14-s + (0.162 + 0.712i)15-s + (−0.988 + 0.149i)16-s + (−0.233 − 0.403i)18-s + (−0.900 + 0.433i)20-s + (0.0546 − 0.728i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.117 - 0.993i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.117 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.795965383\)
\(L(\frac12)\) \(\approx\) \(1.795965383\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.733 - 0.680i)T \)
5 \( 1 + (-0.365 - 0.930i)T \)
7 \( 1 + (0.0747 + 0.997i)T \)
good3 \( 1 + (-0.722 - 0.108i)T + (0.955 + 0.294i)T^{2} \)
11 \( 1 + (-0.826 + 0.563i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.63 + 1.11i)T + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (1.48 - 0.716i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (1.19 + 1.49i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.914 - 0.848i)T + (0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (0.147 - 1.97i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.0747 + 0.997i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.0332 - 0.145i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-1.57 - 0.487i)T + (0.826 + 0.563i)T^{2} \)
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

−10.59255302300550425666497443340, −9.296092267254613746545092558582, −8.710919163387870846996683433529, −7.51252076150191815476583830202, −7.13658811872638273567288839503, −6.20986884731605788703622947891, −5.26021309395698575867664900341, −4.00047323969555791638395892423, −3.30458613928910053006148989174, −2.39648919328569146853149944006, 1.60561369764334737488528690094, 2.59006719863379730046151873032, 3.48992051266533769164603487099, 4.80139838931463354868514016871, 5.48847639823711657946337207096, 6.21334587773823504710467518829, 7.64748466465604297004907222700, 8.637591471516904571430841493530, 9.326941673125162177977065380918, 9.706691113412387146508813059997

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