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Properties

Label 2-980-28.27-c1-0-45
Degree $2$
Conductor $980$
Sign $-0.0773 + 0.997i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 + 0.442i)2-s − 0.901·3-s + (1.60 − 1.18i)4-s − i·5-s + (1.21 − 0.398i)6-s + (−1.63 + 2.30i)8-s − 2.18·9-s + (0.442 + 1.34i)10-s + 3.74i·11-s + (−1.44 + 1.07i)12-s − 2.41i·13-s + 0.901i·15-s + (1.17 − 3.82i)16-s + 0.583i·17-s + (2.93 − 0.967i)18-s + 6.15·19-s + ⋯
L(s)  = 1  + (−0.949 + 0.312i)2-s − 0.520·3-s + (0.804 − 0.594i)4-s − 0.447i·5-s + (0.494 − 0.162i)6-s + (−0.578 + 0.815i)8-s − 0.729·9-s + (0.139 + 0.424i)10-s + 1.12i·11-s + (−0.418 + 0.309i)12-s − 0.671i·13-s + 0.232i·15-s + (0.293 − 0.955i)16-s + 0.141i·17-s + (0.692 − 0.228i)18-s + 1.41·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0773 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0773 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.0773 + 0.997i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.0773 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327187 - 0.353565i\)
\(L(\frac12)\) \(\approx\) \(0.327187 - 0.353565i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.442i)T \)
5 \( 1 + iT \)
7 \( 1 \)
good3 \( 1 + 0.901T + 3T^{2} \)
11 \( 1 - 3.74iT - 11T^{2} \)
13 \( 1 + 2.41iT - 13T^{2} \)
17 \( 1 - 0.583iT - 17T^{2} \)
19 \( 1 - 6.15T + 19T^{2} \)
23 \( 1 + 4.31iT - 23T^{2} \)
29 \( 1 + 0.435T + 29T^{2} \)
31 \( 1 - 2.53T + 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + 7.35iT - 41T^{2} \)
43 \( 1 + 5.80iT - 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 3.11T + 53T^{2} \)
59 \( 1 - 3.47T + 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
67 \( 1 + 9.84iT - 67T^{2} \)
71 \( 1 - 9.96iT - 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 + 0.459iT - 79T^{2} \)
83 \( 1 + 2.59T + 83T^{2} \)
89 \( 1 + 9.88iT - 89T^{2} \)
97 \( 1 + 4.54iT - 97T^{2} \)
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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

−9.817556179519680284953946207419, −8.896791225522249357400337923944, −8.198549754217040371421530291092, −7.31747629765421286380959704523, −6.50954892427541359935705242363, −5.47234094758701315865337870288, −4.93396687169036100051406797646, −3.18299540381328235639311594359, −1.83437628404043453179665608182, −0.35141758459469523762374816199, 1.25187849366145909565766760715, 2.83453274847513984901259634389, 3.53785760250028641231909624145, 5.22036263799560420434979944769, 6.13665570921498613856783777899, 6.88132771167523501727048006105, 7.85012335446375397211927605967, 8.622660064359537180054782148369, 9.443476050342028976105520547215, 10.20768037284709525823450994557

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