Properties

Label 2-1008-252.31-c1-0-26
Degree 22
Conductor 10081008
Sign 0.996+0.0862i0.996 + 0.0862i
Analytic cond. 8.048928.04892
Root an. cond. 2.837062.83706
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  + (1.5 − 0.866i)3-s + 0.717i·5-s + (1.62 + 2.09i)7-s + (1.5 − 2.59i)9-s − 1.73i·11-s + (3.62 + 2.09i)13-s + (0.621 + 1.07i)15-s + (−2.74 − 1.58i)17-s + (0.5 + 0.866i)19-s + (4.24 + 1.73i)21-s + 2.74i·23-s + 4.48·25-s − 5.19i·27-s + (0.621 + 1.07i)29-s + (−2 − 3.46i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + 0.320i·5-s + (0.612 + 0.790i)7-s + (0.5 − 0.866i)9-s − 0.522i·11-s + (1.00 + 0.579i)13-s + (0.160 + 0.277i)15-s + (−0.665 − 0.384i)17-s + (0.114 + 0.198i)19-s + (0.925 + 0.377i)21-s + 0.572i·23-s + 0.897·25-s − 0.999i·27-s + (0.115 + 0.199i)29-s + (−0.359 − 0.622i)31-s + ⋯

Functional equation

Λ(s)=(1008s/2ΓC(s)L(s)=((0.996+0.0862i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2ΓC(s+1/2)L(s)=((0.996+0.0862i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    243272^{4} \cdot 3^{2} \cdot 7
Sign: 0.996+0.0862i0.996 + 0.0862i
Analytic conductor: 8.048928.04892
Root analytic conductor: 2.837062.83706
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1008(31,)\chi_{1008} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :1/2), 0.996+0.0862i)(2,\ 1008,\ (\ :1/2),\ 0.996 + 0.0862i)

Particular Values

L(1)L(1) \approx 2.4158053282.415805328
L(12)L(\frac12) \approx 2.4158053282.415805328
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(1.5+0.866i)T 1 + (-1.5 + 0.866i)T
7 1+(1.622.09i)T 1 + (-1.62 - 2.09i)T
good5 10.717iT5T2 1 - 0.717iT - 5T^{2}
11 1+1.73iT11T2 1 + 1.73iT - 11T^{2}
13 1+(3.622.09i)T+(6.5+11.2i)T2 1 + (-3.62 - 2.09i)T + (6.5 + 11.2i)T^{2}
17 1+(2.74+1.58i)T+(8.5+14.7i)T2 1 + (2.74 + 1.58i)T + (8.5 + 14.7i)T^{2}
19 1+(0.50.866i)T+(9.5+16.4i)T2 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2}
23 12.74iT23T2 1 - 2.74iT - 23T^{2}
29 1+(0.6211.07i)T+(14.5+25.1i)T2 1 + (-0.621 - 1.07i)T + (-14.5 + 25.1i)T^{2}
31 1+(2+3.46i)T+(15.5+26.8i)T2 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.622.80i)T+(18.5+32.0i)T2 1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2}
41 1+(8.745.04i)T+(20.5+35.5i)T2 1 + (-8.74 - 5.04i)T + (20.5 + 35.5i)T^{2}
43 1+(5.743.31i)T+(21.537.2i)T2 1 + (5.74 - 3.31i)T + (21.5 - 37.2i)T^{2}
47 1+(4.24+7.34i)T+(23.540.7i)T2 1 + (-4.24 + 7.34i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.6211.07i)T+(26.545.8i)T2 1 + (0.621 - 1.07i)T + (-26.5 - 45.8i)T^{2}
59 1+(3+5.19i)T+(29.5+51.0i)T2 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2}
61 1+(63.46i)T+(30.5+52.8i)T2 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2}
67 1+(33.558.0i)T2 1 + (33.5 - 58.0i)T^{2}
71 1+13.2iT71T2 1 + 13.2iT - 71T^{2}
73 1+(7.5+4.33i)T+(36.5+63.2i)T2 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2}
79 1+(1.751.01i)T+(39.5+68.4i)T2 1 + (-1.75 - 1.01i)T + (39.5 + 68.4i)T^{2}
83 1+(5.74+9.94i)T+(41.5+71.8i)T2 1 + (5.74 + 9.94i)T + (-41.5 + 71.8i)T^{2}
89 1+(14.28.21i)T+(44.577.0i)T2 1 + (14.2 - 8.21i)T + (44.5 - 77.0i)T^{2}
97 1+(5.74+3.31i)T+(48.584.0i)T2 1 + (-5.74 + 3.31i)T + (48.5 - 84.0i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.686639087677233519138041981441, −8.922435036878865869291952960451, −8.423039171619323585334585156851, −7.55811230050890886774511042675, −6.60763339465528680760884743295, −5.85567880251763762219836403259, −4.57234564297452348214584034609, −3.44849115990989611222559157105, −2.50050965034555916865752689139, −1.39194112199450662851100491356, 1.28909731602658871808620712879, 2.60564634505660515025632565159, 3.87315565694976427759297664122, 4.46520523520664842589658749324, 5.45048618529686864389697415919, 6.80974553855314858820398758567, 7.62298323539855649434383871201, 8.479716613649731953062449461639, 8.946379025948332111986667573245, 10.02845029878535939604319909301

Graph of the ZZ-function along the critical line