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Properties

Label 2-980-140.39-c0-0-2
Degree $2$
Conductor $980$
Sign $0.445 - 0.895i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.965 + 0.258i)5-s + 0.999i·8-s + (0.5 + 0.866i)9-s + (−0.965 + 0.258i)10-s + 1.41i·13-s + (−0.5 − 0.866i)16-s + (−1.22 − 0.707i)17-s + (−0.866 − 0.499i)18-s + (0.707 − 0.707i)20-s + (0.866 + 0.499i)25-s + (−0.707 − 1.22i)26-s + (0.866 + 0.499i)32-s + 1.41·34-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.965 + 0.258i)5-s + 0.999i·8-s + (0.5 + 0.866i)9-s + (−0.965 + 0.258i)10-s + 1.41i·13-s + (−0.5 − 0.866i)16-s + (−1.22 − 0.707i)17-s + (−0.866 − 0.499i)18-s + (0.707 − 0.707i)20-s + (0.866 + 0.499i)25-s + (−0.707 − 1.22i)26-s + (0.866 + 0.499i)32-s + 1.41·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.445 - 0.895i$
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: \((2,\ 980,\ (\ :0),\ 0.445 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8118140285\)
\(L(\frac12)\) \(\approx\) \(0.8118140285\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−10.13350315453744253142215219146, −9.427678038380259738658595103511, −8.898112814802905799253244141719, −7.80425778495939314807713447144, −6.89467643805851499417020845301, −6.43066216532896667696786151439, −5.29460104458426575260161139381, −4.45990399303347121412776320637, −2.46523788384924324392822915304, −1.73813263557611878761106804737, 1.13762375495705468201852172298, 2.38419108559785237991325984166, 3.47096316806423292961315868201, 4.68586867065402267711931142213, 6.05691586499298754202119379166, 6.63751717619188553993042604199, 7.76921478043998931253073608089, 8.623949508148556365066172902088, 9.287796412061601288925450795584, 10.10325307799888726210603182174

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