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Properties

Label 2-980-28.27-c1-0-54
Degree 22
Conductor 980980
Sign 0.972−0.234i0.972 - 0.234i
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  + (−0.431 + 1.34i)2-s + 2.73·3-s + (−1.62 − 1.16i)4-s − i·5-s + (−1.18 + 3.68i)6-s + (2.26 − 1.69i)8-s + 4.49·9-s + (1.34 + 0.431i)10-s + 0.100i·11-s + (−4.45 − 3.18i)12-s − 4.11i·13-s − 2.73i·15-s + (1.29 + 3.78i)16-s − 5.39i·17-s + (−1.93 + 6.05i)18-s + 7.45·19-s + ⋯
L(s)  = 1  + (−0.305 + 0.952i)2-s + 1.58·3-s + (−0.813 − 0.581i)4-s − 0.447i·5-s + (−0.482 + 1.50i)6-s + (0.801 − 0.597i)8-s + 1.49·9-s + (0.425 + 0.136i)10-s + 0.0302i·11-s + (−1.28 − 0.918i)12-s − 1.14i·13-s − 0.706i·15-s + (0.324 + 0.945i)16-s − 1.30i·17-s + (−0.456 + 1.42i)18-s + 1.71·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.972−0.234i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.972−0.234i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.972−0.234i0.972 - 0.234i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(391,⋅)\chi_{980} (391, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.972−0.234i)(2,\ 980,\ (\ :1/2),\ 0.972 - 0.234i)

Particular Values

L(1)L(1) ≈\approx 2.18522+0.260270i2.18522 + 0.260270i
L(12)L(\frac12) ≈\approx 2.18522+0.260270i2.18522 + 0.260270i
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+(0.431−1.34i)T 1 + (0.431 - 1.34i)T
5 1+iT 1 + iT
7 1 1
good3 1−2.73T+3T2 1 - 2.73T + 3T^{2}
11 1−0.100iT−11T2 1 - 0.100iT - 11T^{2}
13 1+4.11iT−13T2 1 + 4.11iT - 13T^{2}
17 1+5.39iT−17T2 1 + 5.39iT - 17T^{2}
19 1−7.45T+19T2 1 - 7.45T + 19T^{2}
23 1+1.50iT−23T2 1 + 1.50iT - 23T^{2}
29 1+2.37T+29T2 1 + 2.37T + 29T^{2}
31 1+5.44T+31T2 1 + 5.44T + 31T^{2}
37 1−1.03T+37T2 1 - 1.03T + 37T^{2}
41 1−7.99iT−41T2 1 - 7.99iT - 41T^{2}
43 1−7.04iT−43T2 1 - 7.04iT - 43T^{2}
47 1−4.44T+47T2 1 - 4.44T + 47T^{2}
53 1−6.14T+53T2 1 - 6.14T + 53T^{2}
59 1−8.52T+59T2 1 - 8.52T + 59T^{2}
61 1−7.90iT−61T2 1 - 7.90iT - 61T^{2}
67 1+0.109iT−67T2 1 + 0.109iT - 67T^{2}
71 1+6.73iT−71T2 1 + 6.73iT - 71T^{2}
73 1−6.14iT−73T2 1 - 6.14iT - 73T^{2}
79 1+4.27iT−79T2 1 + 4.27iT - 79T^{2}
83 1−6.50T+83T2 1 - 6.50T + 83T^{2}
89 1+3.19iT−89T2 1 + 3.19iT - 89T^{2}
97 1−11.0iT−97T2 1 - 11.0iT - 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

−9.483093773043967448584297559871, −9.220188205367784361668475526303, −8.231995326416360431527795167652, −7.68392540116267843211724394670, −7.09568584331847285589311868809, −5.66877906720526979557786213291, −4.89632328300167431726127820670, −3.70247337991162002946968602361, −2.71988592632244488697546733672, −1.06392150944329982998444190262, 1.63215786295818948934965192801, 2.43487672800167087273396131219, 3.60522106835359230482331808353, 3.92042269509667148517289196158, 5.43798910352735752323356525266, 7.06878483829745946742259977001, 7.67346021476486395317565260089, 8.589380426640651138674015531127, 9.179868663020377513593104798631, 9.791479665363594977715563272830

Graph of the ZZ-function along the critical line