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Properties

Label 2-862-431.10-c1-0-1
Degree 22
Conductor 862862
Sign −0.961+0.273i-0.961 + 0.273i
Analytic cond. 6.883106.88310
Root an. cond. 2.623562.62356
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  + (−0.109 + 0.994i)2-s + (−2.97 + 0.437i)3-s + (−0.976 − 0.217i)4-s + (0.296 − 0.0129i)5-s + (−0.109 − 3.00i)6-s + (−1.62 − 2.34i)7-s + (0.322 − 0.946i)8-s + (5.77 − 1.73i)9-s + (−0.0194 + 0.295i)10-s + (3.35 + 5.14i)11-s + (2.99 + 0.219i)12-s + (0.631 + 0.694i)13-s + (2.51 − 1.36i)14-s + (−0.874 + 0.168i)15-s + (0.905 + 0.424i)16-s + (0.141 − 2.76i)17-s + ⋯
L(s)  = 1  + (−0.0773 + 0.702i)2-s + (−1.71 + 0.252i)3-s + (−0.488 − 0.108i)4-s + (0.132 − 0.00580i)5-s + (−0.0448 − 1.22i)6-s + (−0.615 − 0.887i)7-s + (0.114 − 0.334i)8-s + (1.92 − 0.579i)9-s + (−0.00615 + 0.0935i)10-s + (1.01 + 1.55i)11-s + (0.865 + 0.0633i)12-s + (0.175 + 0.192i)13-s + (0.671 − 0.363i)14-s + (−0.225 + 0.0434i)15-s + (0.226 + 0.106i)16-s + (0.0343 − 0.671i)17-s + ⋯

Functional equation

Λ(s)=(862s/2 ΓC(s) L(s)=((−0.961+0.273i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(862s/2 ΓC(s+1/2) L(s)=((−0.961+0.273i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 862862    =    2⋅4312 \cdot 431
Sign: −0.961+0.273i-0.961 + 0.273i
Analytic conductor: 6.883106.88310
Root analytic conductor: 2.623562.62356
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ862(441,⋅)\chi_{862} (441, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 862, ( :1/2), −0.961+0.273i)(2,\ 862,\ (\ :1/2),\ -0.961 + 0.273i)

Particular Values

L(1)L(1) ≈\approx 0.0374466−0.268185i0.0374466 - 0.268185i
L(12)L(\frac12) ≈\approx 0.0374466−0.268185i0.0374466 - 0.268185i
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+(0.109−0.994i)T 1 + (0.109 - 0.994i)T
431 1+(−8.62+18.8i)T 1 + (-8.62 + 18.8i)T
good3 1+(2.97−0.437i)T+(2.87−0.864i)T2 1 + (2.97 - 0.437i)T + (2.87 - 0.864i)T^{2}
5 1+(−0.296+0.0129i)T+(4.98−0.437i)T2 1 + (-0.296 + 0.0129i)T + (4.98 - 0.437i)T^{2}
7 1+(1.62+2.34i)T+(−2.45+6.55i)T2 1 + (1.62 + 2.34i)T + (-2.45 + 6.55i)T^{2}
11 1+(−3.35−5.14i)T+(−4.44+10.0i)T2 1 + (-3.35 - 5.14i)T + (-4.44 + 10.0i)T^{2}
13 1+(−0.631−0.694i)T+(−1.23+12.9i)T2 1 + (-0.631 - 0.694i)T + (-1.23 + 12.9i)T^{2}
17 1+(−0.141+2.76i)T+(−16.9−1.73i)T2 1 + (-0.141 + 2.76i)T + (-16.9 - 1.73i)T^{2}
19 1+(−1.09−3.94i)T+(−16.2+9.77i)T2 1 + (-1.09 - 3.94i)T + (-16.2 + 9.77i)T^{2}
23 1+(4.48+1.06i)T+(20.5+10.3i)T2 1 + (4.48 + 1.06i)T + (20.5 + 10.3i)T^{2}
29 1+(−4.13−1.79i)T+(19.8+21.1i)T2 1 + (-4.13 - 1.79i)T + (19.8 + 21.1i)T^{2}
31 1+(1.66+0.609i)T+(23.6+20.0i)T2 1 + (1.66 + 0.609i)T + (23.6 + 20.0i)T^{2}
37 1+(5.02+4.76i)T+(1.89+36.9i)T2 1 + (5.02 + 4.76i)T + (1.89 + 36.9i)T^{2}
41 1+(−5.00+5.34i)T+(−2.69−40.9i)T2 1 + (-5.00 + 5.34i)T + (-2.69 - 40.9i)T^{2}
43 1+(−0.447−6.78i)T+(−42.6+5.63i)T2 1 + (-0.447 - 6.78i)T + (-42.6 + 5.63i)T^{2}
47 1+(3.45−4.83i)T+(−15.1−44.4i)T2 1 + (3.45 - 4.83i)T + (-15.1 - 44.4i)T^{2}
53 1+(0.243−0.329i)T+(−15.6−50.6i)T2 1 + (0.243 - 0.329i)T + (-15.6 - 50.6i)T^{2}
59 1+(6.58+2.97i)T+(39.0+44.2i)T2 1 + (6.58 + 2.97i)T + (39.0 + 44.2i)T^{2}
61 1+(8.14−7.51i)T+(4.89−60.8i)T2 1 + (8.14 - 7.51i)T + (4.89 - 60.8i)T^{2}
67 1+(1.14−2.40i)T+(−42.1−52.1i)T2 1 + (1.14 - 2.40i)T + (-42.1 - 52.1i)T^{2}
71 1+(−0.0423−5.79i)T+(−70.9+1.03i)T2 1 + (-0.0423 - 5.79i)T + (-70.9 + 1.03i)T^{2}
73 1+(−2.36+1.51i)T+(30.5−66.3i)T2 1 + (-2.36 + 1.51i)T + (30.5 - 66.3i)T^{2}
79 1+(11.9−0.348i)T+(78.8−4.61i)T2 1 + (11.9 - 0.348i)T + (78.8 - 4.61i)T^{2}
83 1+(9.34+4.89i)T+(47.3+68.2i)T2 1 + (9.34 + 4.89i)T + (47.3 + 68.2i)T^{2}
89 1+(15.3−7.48i)T+(54.9−70.0i)T2 1 + (15.3 - 7.48i)T + (54.9 - 70.0i)T^{2}
97 1+(−4.23−14.4i)T+(−81.6+52.3i)T2 1 + (-4.23 - 14.4i)T + (-81.6 + 52.3i)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

−10.38254013821199732943633078482, −9.918190384134668884961633005661, −9.278483621684623925729581974499, −7.62220626545836850385709803647, −6.98020255529590939010809384289, −6.33430604089525718663833159437, −5.57006392396467366968726165468, −4.45424493797456560357096498030, −3.97415671160711269718192477600, −1.39301200231786700591668220429, 0.18886724188691429136947675335, 1.52392287741636997127046471332, 3.15212721396241373883272314495, 4.30565333943694683830348553896, 5.58629104157893489621421478335, 5.99956300500990688879327338247, 6.71591557793335667205666693350, 8.172110103270191765504596260896, 9.078140655760143259621691814568, 9.943946334909771852643666130051

Graph of the ZZ-function along the critical line