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Properties

Label 2-980-140.39-c0-0-3
Degree $2$
Conductor $980$
Sign $0.386 - 0.922i$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·6-s − 0.999·8-s + (−0.499 + 0.866i)10-s + (0.499 + 0.866i)12-s + 0.999·15-s + (−0.5 − 0.866i)16-s − 0.999·20-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)24-s + (−0.499 + 0.866i)25-s + 27-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·6-s − 0.999·8-s + (−0.499 + 0.866i)10-s + (0.499 + 0.866i)12-s + 0.999·15-s + (−0.5 − 0.866i)16-s − 0.999·20-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)24-s + (−0.499 + 0.866i)25-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.518328837\)
\(L(\frac12)\) \(\approx\) \(1.518328837\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−10.27063485513183203739136769420, −9.306118320026644146129656688557, −8.386841777392394641761873334448, −7.61834630716422648512255728174, −6.99710191533280619868720604093, −6.28411092865483912116435593223, −5.42034426440017055741925048047, −4.17212403703495731685669257476, −3.01324797728851002423687673662, −2.07868888227772951163696460029, 1.44552514691148156794630113023, 2.75032645736142701288441007675, 3.86209325741397357243344498892, 4.50244504137645818918700393510, 5.43849516585984197964514009485, 6.23179800747396393729281022283, 7.78613230860870997643551956293, 8.920743982118600142009791349902, 9.367407458892367353981578338808, 9.927437613709668642929250081885

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