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Properties

Label 2-10e3-1000.109-c1-0-104
Degree 22
Conductor 10001000
Sign 0.994+0.104i0.994 + 0.104i
Analytic cond. 7.985047.98504
Root an. cond. 2.825782.82578
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  + (1.35 − 0.390i)2-s + (0.290 + 0.351i)3-s + (1.69 − 1.06i)4-s + (1.00 + 1.99i)5-s + (0.532 + 0.364i)6-s + (3.27 − 1.06i)7-s + (1.88 − 2.10i)8-s + (0.523 − 2.74i)9-s + (2.15 + 2.31i)10-s + (−0.505 + 4.00i)11-s + (0.866 + 0.287i)12-s + (−0.348 + 1.82i)13-s + (4.03 − 2.72i)14-s + (−0.407 + 0.935i)15-s + (1.74 − 3.59i)16-s + (−0.351 − 1.36i)17-s + ⋯
L(s)  = 1  + (0.961 − 0.276i)2-s + (0.167 + 0.203i)3-s + (0.847 − 0.530i)4-s + (0.451 + 0.892i)5-s + (0.217 + 0.148i)6-s + (1.23 − 0.402i)7-s + (0.667 − 0.744i)8-s + (0.174 − 0.914i)9-s + (0.680 + 0.732i)10-s + (−0.152 + 1.20i)11-s + (0.250 + 0.0828i)12-s + (−0.0967 + 0.507i)13-s + (1.07 − 0.729i)14-s + (−0.105 + 0.241i)15-s + (0.436 − 0.899i)16-s + (−0.0851 − 0.331i)17-s + ⋯

Functional equation

Λ(s)=(1000s/2 ΓC(s) L(s)=((0.994+0.104i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1000s/2 ΓC(s+1/2) L(s)=((0.994+0.104i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: 0.994+0.104i0.994 + 0.104i
Analytic conductor: 7.985047.98504
Root analytic conductor: 2.825782.82578
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1000(109,⋅)\chi_{1000} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1000, ( :1/2), 0.994+0.104i)(2,\ 1000,\ (\ :1/2),\ 0.994 + 0.104i)

Particular Values

L(1)L(1) ≈\approx 3.69051−0.193365i3.69051 - 0.193365i
L(12)L(\frac12) ≈\approx 3.69051−0.193365i3.69051 - 0.193365i
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+(−1.35+0.390i)T 1 + (-1.35 + 0.390i)T
5 1+(−1.00−1.99i)T 1 + (-1.00 - 1.99i)T
good3 1+(−0.290−0.351i)T+(−0.562+2.94i)T2 1 + (-0.290 - 0.351i)T + (-0.562 + 2.94i)T^{2}
7 1+(−3.27+1.06i)T+(5.66−4.11i)T2 1 + (-3.27 + 1.06i)T + (5.66 - 4.11i)T^{2}
11 1+(0.505−4.00i)T+(−10.6−2.73i)T2 1 + (0.505 - 4.00i)T + (-10.6 - 2.73i)T^{2}
13 1+(0.348−1.82i)T+(−12.0−4.78i)T2 1 + (0.348 - 1.82i)T + (-12.0 - 4.78i)T^{2}
17 1+(0.351+1.36i)T+(−14.8+8.18i)T2 1 + (0.351 + 1.36i)T + (-14.8 + 8.18i)T^{2}
19 1+(4.46+3.69i)T+(3.56+18.6i)T2 1 + (4.46 + 3.69i)T + (3.56 + 18.6i)T^{2}
23 1+(2.55−2.72i)T+(−1.44−22.9i)T2 1 + (2.55 - 2.72i)T + (-1.44 - 22.9i)T^{2}
29 1+(8.15−0.513i)T+(28.7−3.63i)T2 1 + (8.15 - 0.513i)T + (28.7 - 3.63i)T^{2}
31 1+(−5.40+1.38i)T+(27.1−14.9i)T2 1 + (-5.40 + 1.38i)T + (27.1 - 14.9i)T^{2}
37 1+(−4.18−2.30i)T+(19.8+31.2i)T2 1 + (-4.18 - 2.30i)T + (19.8 + 31.2i)T^{2}
41 1+(2.71−2.55i)T+(2.57−40.9i)T2 1 + (2.71 - 2.55i)T + (2.57 - 40.9i)T^{2}
43 1+(0.804−0.584i)T+(13.2−40.8i)T2 1 + (0.804 - 0.584i)T + (13.2 - 40.8i)T^{2}
47 1+(2.83+7.16i)T+(−34.2+32.1i)T2 1 + (2.83 + 7.16i)T + (-34.2 + 32.1i)T^{2}
53 1+(0.588−0.928i)T+(−22.5−47.9i)T2 1 + (0.588 - 0.928i)T + (-22.5 - 47.9i)T^{2}
59 1+(−6.95+3.27i)T+(37.6−45.4i)T2 1 + (-6.95 + 3.27i)T + (37.6 - 45.4i)T^{2}
61 1+(−4.68+4.98i)T+(−3.83−60.8i)T2 1 + (-4.68 + 4.98i)T + (-3.83 - 60.8i)T^{2}
67 1+(0.294−4.68i)T+(−66.4−8.39i)T2 1 + (0.294 - 4.68i)T + (-66.4 - 8.39i)T^{2}
71 1+(0.689−0.272i)T+(51.7−48.6i)T2 1 + (0.689 - 0.272i)T + (51.7 - 48.6i)T^{2}
73 1+(−10.0−4.71i)T+(46.5+56.2i)T2 1 + (-10.0 - 4.71i)T + (46.5 + 56.2i)T^{2}
79 1+(−1.98−2.39i)T+(−14.8+77.6i)T2 1 + (-1.98 - 2.39i)T + (-14.8 + 77.6i)T^{2}
83 1+(7.58−9.17i)T+(−15.5−81.5i)T2 1 + (7.58 - 9.17i)T + (-15.5 - 81.5i)T^{2}
89 1+(5.18−11.0i)T+(−56.7−68.5i)T2 1 + (5.18 - 11.0i)T + (-56.7 - 68.5i)T^{2}
97 1+(15.5−0.979i)T+(96.2−12.1i)T2 1 + (15.5 - 0.979i)T + (96.2 - 12.1i)T^{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

−9.920636861673252069338650157312, −9.619138393458836664334774445512, −8.119007330585495876855219412901, −7.02386482116754883050892941261, −6.70975753130965358560655998595, −5.47155423003557253181456071091, −4.51260159855486815928478559736, −3.86219499188602296694269930052, −2.52113917617896729264078295140, −1.67987101090665534995700565782, 1.63337242325594792491838685166, 2.45875845395848842211350674411, 3.98731969900334229284631838116, 4.87227655334763947664478227999, 5.57539097919316991024421843568, 6.20742932451619485162669783281, 7.71007628723657194494630345624, 8.215190690050056385854052176729, 8.678386259077304992298819466113, 10.24495555548416155219588972766

Graph of the ZZ-function along the critical line