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Properties

Label 2-980-140.3-c0-0-0
Degree $2$
Conductor $980$
Sign $-0.442 - 0.896i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.793 − 0.608i)5-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.608 + 0.793i)10-s + (−1.30 + 1.30i)13-s + (0.500 + 0.866i)16-s + (−0.739 + 0.198i)17-s + (0.965 − 0.258i)18-s + (−0.382 − 0.923i)20-s + (0.258 + 0.965i)25-s + (1.60 − 0.923i)26-s + 1.41i·29-s + (−0.258 − 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.793 − 0.608i)5-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.608 + 0.793i)10-s + (−1.30 + 1.30i)13-s + (0.500 + 0.866i)16-s + (−0.739 + 0.198i)17-s + (0.965 − 0.258i)18-s + (−0.382 − 0.923i)20-s + (0.258 + 0.965i)25-s + (1.60 − 0.923i)26-s + 1.41i·29-s + (−0.258 − 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.442 - 0.896i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ -0.442 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2098698085\)
\(L(\frac12)\) \(\approx\) \(0.2098698085\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (0.793 + 0.608i)T \)
7 \( 1 \)
good3 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
17 \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 + 0.765iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
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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

−10.58509331467899059631913064337, −9.273059632179850819792288147242, −9.007749242780114431572499046949, −8.071453204772221924978167558190, −7.36070479481897925839917517709, −6.54970014253360384761863622464, −5.18747970343789059485017474280, −4.22979738489037623592835332475, −2.94868476808694279047656894752, −1.80173767710441374026707575090, 0.25224155225769042694524968774, 2.47895911302942702865657782955, 3.21953592942144381038144394182, 4.77912142873799789323165718452, 5.93274768787932598122853440615, 6.69051636225399937448884141736, 7.70118326770125708436697465717, 8.056571670827168070611320271409, 9.093945163117925026122343967201, 9.903195382520883118680170589129

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