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Properties

Label 2-980-140.3-c0-0-1
Degree $2$
Conductor $980$
Sign $0.981 - 0.193i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.981 - 0.193i$
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: \((2,\ 980,\ (\ :0),\ 0.981 - 0.193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7707417699\)
\(L(\frac12)\) \(\approx\) \(0.7707417699\)
\(L(1)\) 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

−10.26797414631859310701165028802, −9.485943152340309913810841014007, −8.530561981723582105322901514962, −7.984916823979317405085786461897, −6.95328766250219638682838396168, −6.01780805418198657505231114166, −5.38272519424650650531268681142, −3.38371577341425762399851584023, −2.81285987596997729614699326455, −1.43215556003379197628671877211, 1.22572677767808113077800538271, 2.38780350328171679006700652100, 3.84035609349990393963401059427, 5.32863367176971481659335608254, 6.11270401451517056302633784411, 6.62766314721028818702805191154, 7.973510121635309259561544578837, 8.652559839790312373250581741753, 9.259255153303931927987753291837, 9.886553975022298549578761837889

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