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Properties

Label 2-2070-69.68-c1-0-18
Degree 22
Conductor 20702070
Sign −0.619+0.785i-0.619 + 0.785i
Analytic cond. 16.529016.5290
Root an. cond. 4.065594.06559
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  − i·2-s − 4-s − 5-s + 2.68i·7-s + i·8-s + i·10-s − 1.73·11-s − 2.55·13-s + 2.68·14-s + 16-s − 1.27·17-s + 1.40i·19-s + 20-s + 1.73i·22-s + (4.59 − 1.35i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.447·5-s + 1.01i·7-s + 0.353i·8-s + 0.316i·10-s − 0.521·11-s − 0.708·13-s + 0.718·14-s + 0.250·16-s − 0.308·17-s + 0.322i·19-s + 0.223·20-s + 0.369i·22-s + (0.958 − 0.283i)23-s + ⋯

Functional equation

Λ(s)=(2070s/2 ΓC(s) L(s)=((−0.619+0.785i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2070s/2 ΓC(s+1/2) L(s)=((−0.619+0.785i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20702070    =    2⋅32⋅5⋅232 \cdot 3^{2} \cdot 5 \cdot 23
Sign: −0.619+0.785i-0.619 + 0.785i
Analytic conductor: 16.529016.5290
Root analytic conductor: 4.065594.06559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2070(1241,⋅)\chi_{2070} (1241, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2070, ( :1/2), −0.619+0.785i)(2,\ 2070,\ (\ :1/2),\ -0.619 + 0.785i)

Particular Values

L(1)L(1) ≈\approx 0.82438845910.8243884591
L(12)L(\frac12) ≈\approx 0.82438845910.8243884591
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+iT 1 + iT
3 1 1
5 1+T 1 + T
23 1+(−4.59+1.35i)T 1 + (-4.59 + 1.35i)T
good7 1−2.68iT−7T2 1 - 2.68iT - 7T^{2}
11 1+1.73T+11T2 1 + 1.73T + 11T^{2}
13 1+2.55T+13T2 1 + 2.55T + 13T^{2}
17 1+1.27T+17T2 1 + 1.27T + 17T^{2}
19 1−1.40iT−19T2 1 - 1.40iT - 19T^{2}
29 1+9.62iT−29T2 1 + 9.62iT - 29T^{2}
31 1+1.47T+31T2 1 + 1.47T + 31T^{2}
37 1+4.20iT−37T2 1 + 4.20iT - 37T^{2}
41 1+3.40iT−41T2 1 + 3.40iT - 41T^{2}
43 1−2.35iT−43T2 1 - 2.35iT - 43T^{2}
47 1+3.21iT−47T2 1 + 3.21iT - 47T^{2}
53 1−1.80T+53T2 1 - 1.80T + 53T^{2}
59 1+7.74iT−59T2 1 + 7.74iT - 59T^{2}
61 1+6.78iT−61T2 1 + 6.78iT - 61T^{2}
67 1+7.71iT−67T2 1 + 7.71iT - 67T^{2}
71 1+4.44iT−71T2 1 + 4.44iT - 71T^{2}
73 1+1.36T+73T2 1 + 1.36T + 73T^{2}
79 1−10.0iT−79T2 1 - 10.0iT - 79T^{2}
83 1−8.67T+83T2 1 - 8.67T + 83T^{2}
89 1−3.72T+89T2 1 - 3.72T + 89T^{2}
97 1+7.03iT−97T2 1 + 7.03iT - 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

−8.970460618975119731050282582210, −8.171800645661346691863371833430, −7.49917659552998032091651841237, −6.40250277794341887783079884433, −5.44587165138197125862276279951, −4.77390078899928159523150145495, −3.78063091105328628228840353722, −2.72993466009666545716081346572, −2.06521002000647397402969216571, −0.33500430293841913534665684759, 1.08865995719968532047082567625, 2.80407973642384884818908420701, 3.78747899586088298592979379198, 4.71747431346302441527536917356, 5.26671455295819119314011740618, 6.47569229936701237070878237267, 7.28671094252363425621484348700, 7.49897909369108070677518572086, 8.574587864806215716728625658367, 9.182864228226533680850596854746

Graph of the ZZ-function along the critical line