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Properties

Label 2-980-140.139-c1-0-11
Degree 22
Conductor 980980
Sign −0.995+0.0989i-0.995 + 0.0989i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.995+0.0989i) Λ‾(2−s)\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}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.995+0.0989i) Λ‾(1−s)\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: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.995+0.0989i-0.995 + 0.0989i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(979,⋅)\chi_{980} (979, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.995+0.0989i)(2,\ 980,\ (\ :1/2),\ -0.995 + 0.0989i)

Particular Values

L(1)L(1) ≈\approx 0.0198866−0.400831i0.0198866 - 0.400831i
L(12)L(\frac12) ≈\approx 0.0198866−0.400831i0.0198866 - 0.400831i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.38+0.295i)T 1 + (1.38 + 0.295i)T
5 1+(2.22−0.220i)T 1 + (2.22 - 0.220i)T
7 1 1
good3 1−1.66iT−3T2 1 - 1.66iT - 3T^{2}
11 1−4.81iT−11T2 1 - 4.81iT - 11T^{2}
13 1−2.14T+13T2 1 - 2.14T + 13T^{2}
17 1+5.02T+17T2 1 + 5.02T + 17T^{2}
19 1+1.36T+19T2 1 + 1.36T + 19T^{2}
23 1−5.18T+23T2 1 - 5.18T + 23T^{2}
29 1+6.43T+29T2 1 + 6.43T + 29T^{2}
31 1+4.62T+31T2 1 + 4.62T + 31T^{2}
37 1−9.82iT−37T2 1 - 9.82iT - 37T^{2}
41 1+4.71iT−41T2 1 + 4.71iT - 41T^{2}
43 1−0.141T+43T2 1 - 0.141T + 43T^{2}
47 1−2.55iT−47T2 1 - 2.55iT - 47T^{2}
53 1+4.84iT−53T2 1 + 4.84iT - 53T^{2}
59 1+14.1T+59T2 1 + 14.1T + 59T^{2}
61 1+10.1iT−61T2 1 + 10.1iT - 61T^{2}
67 1+9.64T+67T2 1 + 9.64T + 67T^{2}
71 1+9.58iT−71T2 1 + 9.58iT - 71T^{2}
73 1−1.67T+73T2 1 - 1.67T + 73T^{2}
79 1+11.8iT−79T2 1 + 11.8iT - 79T^{2}
83 1−0.811iT−83T2 1 - 0.811iT - 83T^{2}
89 1−16.0iT−89T2 1 - 16.0iT - 89T^{2}
97 1+1.76T+97T2 1 + 1.76T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line