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Properties

Label 2-980-140.139-c1-0-97
Degree 22
Conductor 980980
Sign −0.563+0.826i-0.563 + 0.826i
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  + (0.576 + 1.29i)2-s − 2.50i·3-s + (−1.33 + 1.48i)4-s + (−0.639 − 2.14i)5-s + (3.23 − 1.44i)6-s + (−2.69 − 0.867i)8-s − 3.25·9-s + (2.39 − 2.06i)10-s − 2.25i·11-s + (3.72 + 3.34i)12-s + 5.96·13-s + (−5.35 + 1.60i)15-s + (−0.430 − 3.97i)16-s − 2.00·17-s + (−1.87 − 4.20i)18-s − 7.81·19-s + ⋯
L(s)  = 1  + (0.407 + 0.913i)2-s − 1.44i·3-s + (−0.667 + 0.744i)4-s + (−0.286 − 0.958i)5-s + (1.31 − 0.588i)6-s + (−0.951 − 0.306i)8-s − 1.08·9-s + (0.758 − 0.651i)10-s − 0.678i·11-s + (1.07 + 0.964i)12-s + 1.65·13-s + (−1.38 + 0.413i)15-s + (−0.107 − 0.994i)16-s − 0.486·17-s + (−0.442 − 0.991i)18-s − 1.79·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.563+0.826i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.563+0.826i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.563+0.826i-0.563 + 0.826i
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), −0.563+0.826i)(2,\ 980,\ (\ :1/2),\ -0.563 + 0.826i)

Particular Values

L(1)L(1) ≈\approx 0.513294−0.971196i0.513294 - 0.971196i
L(12)L(\frac12) ≈\approx 0.513294−0.971196i0.513294 - 0.971196i
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+(−0.576−1.29i)T 1 + (-0.576 - 1.29i)T
5 1+(0.639+2.14i)T 1 + (0.639 + 2.14i)T
7 1 1
good3 1+2.50iT−3T2 1 + 2.50iT - 3T^{2}
11 1+2.25iT−11T2 1 + 2.25iT - 11T^{2}
13 1−5.96T+13T2 1 - 5.96T + 13T^{2}
17 1+2.00T+17T2 1 + 2.00T + 17T^{2}
19 1+7.81T+19T2 1 + 7.81T + 19T^{2}
23 1+2.99T+23T2 1 + 2.99T + 23T^{2}
29 1+4.87T+29T2 1 + 4.87T + 29T^{2}
31 1−1.49T+31T2 1 - 1.49T + 31T^{2}
37 1−4.78iT−37T2 1 - 4.78iT - 37T^{2}
41 1+8.82iT−41T2 1 + 8.82iT - 41T^{2}
43 1−1.12T+43T2 1 - 1.12T + 43T^{2}
47 1+9.56iT−47T2 1 + 9.56iT - 47T^{2}
53 1−7.06iT−53T2 1 - 7.06iT - 53T^{2}
59 1+11.4T+59T2 1 + 11.4T + 59T^{2}
61 1−1.21iT−61T2 1 - 1.21iT - 61T^{2}
67 1−1.11T+67T2 1 - 1.11T + 67T^{2}
71 1+8.40iT−71T2 1 + 8.40iT - 71T^{2}
73 1−5.88T+73T2 1 - 5.88T + 73T^{2}
79 1+12.1iT−79T2 1 + 12.1iT - 79T^{2}
83 1−11.1iT−83T2 1 - 11.1iT - 83T^{2}
89 1+4.57iT−89T2 1 + 4.57iT - 89T^{2}
97 1−4.62T+97T2 1 - 4.62T + 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

−9.086237290280560096205240342783, −8.486161675275840401314420590718, −8.110022771448578019967839900690, −7.11869203574237884642899007496, −6.19306723121130080332764087099, −5.84054866289690574004259330790, −4.46888379026270986908809972769, −3.58153991434028228025157694986, −1.87621388459183671195776340888, −0.43466425106079064757259365859, 2.04795339327876747635332576480, 3.26912688452491026137182428282, 4.04906614969015281873735539230, 4.51015311967516595580254122778, 5.85737806572877580613530695699, 6.53083254216938357005322450035, 8.111044122814674633205098000043, 8.984597525063105224484870990876, 9.745465282309345429218557722985, 10.48340064459453319848845668038

Graph of the ZZ-function along the critical line