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Properties

Label 2-980-140.139-c1-0-83
Degree 22
Conductor 980980
Sign 0.877+0.480i0.877 + 0.480i
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  + (1.36 − 0.385i)2-s + 0.423i·3-s + (1.70 − 1.04i)4-s + (1.74 − 1.39i)5-s + (0.163 + 0.576i)6-s + (1.91 − 2.08i)8-s + 2.82·9-s + (1.83 − 2.57i)10-s + 4.89i·11-s + (0.443 + 0.721i)12-s + 2.54·13-s + (0.591 + 0.738i)15-s + (1.80 − 3.57i)16-s − 5.11·17-s + (3.83 − 1.08i)18-s − 6.26·19-s + ⋯
L(s)  = 1  + (0.962 − 0.272i)2-s + 0.244i·3-s + (0.851 − 0.524i)4-s + (0.780 − 0.625i)5-s + (0.0665 + 0.235i)6-s + (0.676 − 0.736i)8-s + 0.940·9-s + (0.580 − 0.814i)10-s + 1.47i·11-s + (0.128 + 0.208i)12-s + 0.706·13-s + (0.152 + 0.190i)15-s + (0.450 − 0.892i)16-s − 1.24·17-s + (0.904 − 0.256i)18-s − 1.43·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.877+0.480i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.877+0.480i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.877+0.480i0.877 + 0.480i
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), 0.877+0.480i)(2,\ 980,\ (\ :1/2),\ 0.877 + 0.480i)

Particular Values

L(1)L(1) ≈\approx 3.40829−0.871933i3.40829 - 0.871933i
L(12)L(\frac12) ≈\approx 3.40829−0.871933i3.40829 - 0.871933i
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+(−1.36+0.385i)T 1 + (-1.36 + 0.385i)T
5 1+(−1.74+1.39i)T 1 + (-1.74 + 1.39i)T
7 1 1
good3 1−0.423iT−3T2 1 - 0.423iT - 3T^{2}
11 1−4.89iT−11T2 1 - 4.89iT - 11T^{2}
13 1−2.54T+13T2 1 - 2.54T + 13T^{2}
17 1+5.11T+17T2 1 + 5.11T + 17T^{2}
19 1+6.26T+19T2 1 + 6.26T + 19T^{2}
23 1−4.63T+23T2 1 - 4.63T + 23T^{2}
29 1+1.88T+29T2 1 + 1.88T + 29T^{2}
31 1−1.47T+31T2 1 - 1.47T + 31T^{2}
37 1+2.35iT−37T2 1 + 2.35iT - 37T^{2}
41 1−7.05iT−41T2 1 - 7.05iT - 41T^{2}
43 1+10.7T+43T2 1 + 10.7T + 43T^{2}
47 1+12.2iT−47T2 1 + 12.2iT - 47T^{2}
53 1+2.23iT−53T2 1 + 2.23iT - 53T^{2}
59 1−5.68T+59T2 1 - 5.68T + 59T^{2}
61 1+3.87iT−61T2 1 + 3.87iT - 61T^{2}
67 1−0.889T+67T2 1 - 0.889T + 67T^{2}
71 1−14.3iT−71T2 1 - 14.3iT - 71T^{2}
73 1+7.87T+73T2 1 + 7.87T + 73T^{2}
79 1−4.63iT−79T2 1 - 4.63iT - 79T^{2}
83 1−4.32iT−83T2 1 - 4.32iT - 83T^{2}
89 1−2.18iT−89T2 1 - 2.18iT - 89T^{2}
97 1−3.42T+97T2 1 - 3.42T + 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

−10.08144331289907090170611278297, −9.356949741947129691416287042390, −8.361534572258830552888583528444, −6.86957465747803304222051951321, −6.62304856897954652618755288538, −5.29823426016544457168380586580, −4.59199995141487616872482815481, −3.97332543248561962126757533927, −2.31675943873516435726362454523, −1.55321676568007244344693241636, 1.66678782253802865080204071754, 2.79309014095342678315071389019, 3.80075058144729022006812736136, 4.82853127387097897905600753542, 6.02707118301786514825997378751, 6.43436844308908787576074064281, 7.17251964365699594897057207787, 8.330484085943272184587698233461, 9.072158111781253152280009490405, 10.50357826956675770466078578168

Graph of the ZZ-function along the critical line