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Properties

Label 2-980-140.139-c1-0-11
Degree $2$
Conductor $980$
Sign $-0.995 + 0.0989i$
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  + (−1.38 − 0.295i)2-s + 1.66i·3-s + (1.82 + 0.817i)4-s + (−2.22 + 0.220i)5-s + (0.491 − 2.29i)6-s + (−2.28 − 1.67i)8-s + 0.236·9-s + (3.14 + 0.352i)10-s + 4.81i·11-s + (−1.35 + 3.03i)12-s + 2.14·13-s + (−0.366 − 3.69i)15-s + (2.66 + 2.98i)16-s − 5.02·17-s + (−0.326 − 0.0698i)18-s − 1.36·19-s + ⋯
L(s)  = 1  + (−0.977 − 0.209i)2-s + 0.959i·3-s + (0.912 + 0.408i)4-s + (−0.995 + 0.0986i)5-s + (0.200 − 0.938i)6-s + (−0.807 − 0.590i)8-s + 0.0787·9-s + (0.993 + 0.111i)10-s + 1.45i·11-s + (−0.392 + 0.875i)12-s + 0.594·13-s + (−0.0946 − 0.955i)15-s + (0.665 + 0.746i)16-s − 1.21·17-s + (−0.0770 − 0.0164i)18-s − 0.312·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.995 + 0.0989i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.995 + 0.0989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0198866 - 0.400831i\)
\(L(\frac12)\) \(\approx\) \(0.0198866 - 0.400831i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.295i)T \)
5 \( 1 + (2.22 - 0.220i)T \)
7 \( 1 \)
good3 \( 1 - 1.66iT - 3T^{2} \)
11 \( 1 - 4.81iT - 11T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
17 \( 1 + 5.02T + 17T^{2} \)
19 \( 1 + 1.36T + 19T^{2} \)
23 \( 1 - 5.18T + 23T^{2} \)
29 \( 1 + 6.43T + 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 - 9.82iT - 37T^{2} \)
41 \( 1 + 4.71iT - 41T^{2} \)
43 \( 1 - 0.141T + 43T^{2} \)
47 \( 1 - 2.55iT - 47T^{2} \)
53 \( 1 + 4.84iT - 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 + 9.64T + 67T^{2} \)
71 \( 1 + 9.58iT - 71T^{2} \)
73 \( 1 - 1.67T + 73T^{2} \)
79 \( 1 + 11.8iT - 79T^{2} \)
83 \( 1 - 0.811iT - 83T^{2} \)
89 \( 1 - 16.0iT - 89T^{2} \)
97 \( 1 + 1.76T + 97T^{2} \)
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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

−10.60785825319632715959995823610, −9.389010361605334158130633325315, −9.112054934389495666933871125656, −8.023310062190464036608772675749, −7.21399869064591254960395894195, −6.57149170935285815425654207266, −4.91356743556860244475132712672, −4.13032525886699129305797385092, −3.22329661458723348716921015222, −1.76352077327266709218106966521, 0.26467672319380074929214436852, 1.43372749193532313619193647455, 2.83663777893120981612315790363, 4.05071693851969093793922585421, 5.61259644745489784291225764704, 6.47995279449893475011212910203, 7.24330280404845286864813134820, 7.83977504887078179883223081239, 8.759703137117477799470256183016, 9.082351497311752068752666191649

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