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Properties

Label 2-980-140.139-c1-0-6
Degree 22
Conductor 980980
Sign −0.924−0.381i-0.924 - 0.381i
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 + (−3.13 − 0.408i)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 + (−0.991 − 0.129i)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)=((−0.924−0.381i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.924−0.381i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.924−0.381i-0.924 - 0.381i
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), −0.924−0.381i)(2,\ 980,\ (\ :1/2),\ -0.924 - 0.381i)

Particular Values

L(1)L(1) ≈\approx 0.0649546+0.327196i0.0649546 + 0.327196i
L(12)L(\frac12) ≈\approx 0.0649546+0.327196i0.0649546 + 0.327196i
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

−10.12638915348308947898702682423, −9.531300496666343458793194892116, −8.316689187685849304286191507887, −7.60879145320222259471484995169, −6.96358829423518498647533905204, −6.55869829931409425640953062101, −5.57923770750810818217021181556, −4.44493138813143213237126140188, −2.62495151730960501573448190463, −1.68526772990729622009536009751, 0.16772313963424912985265124684, 2.05472960792747094235758711030, 3.27918307978313669021068826438, 4.34074343866254971839446625842, 4.82462335115564839480684767926, 5.76353021205831304842404040753, 7.45675062255404347206829313670, 8.462179886903317164411041042029, 9.157421302963026057935531984299, 9.584754198303044245688150644685

Graph of the ZZ-function along the critical line