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Properties

Label 2-1071-7.4-c1-0-22
Degree 22
Conductor 10711071
Sign 0.386−0.922i0.386 - 0.922i
Analytic cond. 8.551978.55197
Root an. cond. 2.924372.92437
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + (1.5 + 2.59i)5-s + (−2 + 1.73i)7-s + 3·8-s + (−1.5 + 2.59i)10-s + (3 − 5.19i)11-s + 13-s + (−2.5 − 0.866i)14-s + (0.500 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (2 + 3.46i)19-s + 3·20-s + 6·22-s + (2 + 3.46i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.755 + 0.654i)7-s + 1.06·8-s + (−0.474 + 0.821i)10-s + (0.904 − 1.56i)11-s + 0.277·13-s + (−0.668 − 0.231i)14-s + (0.125 + 0.216i)16-s + (−0.121 + 0.210i)17-s + (0.458 + 0.794i)19-s + 0.670·20-s + 1.27·22-s + (0.417 + 0.722i)23-s + ⋯

Functional equation

Λ(s)=(1071s/2 ΓC(s) L(s)=((0.386−0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1071s/2 ΓC(s+1/2) L(s)=((0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10711071    =    32⋅7⋅173^{2} \cdot 7 \cdot 17
Sign: 0.386−0.922i0.386 - 0.922i
Analytic conductor: 8.551978.55197
Root analytic conductor: 2.924372.92437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1071(613,⋅)\chi_{1071} (613, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1071, ( :1/2), 0.386−0.922i)(2,\ 1071,\ (\ :1/2),\ 0.386 - 0.922i)

Particular Values

L(1)L(1) ≈\approx 2.4793210562.479321056
L(12)L(\frac12) ≈\approx 2.4793210562.479321056
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)
bad3 1 1
7 1+(2−1.73i)T 1 + (2 - 1.73i)T
17 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good2 1+(−0.5−0.866i)T+(−1+1.73i)T2 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2}
5 1+(−1.5−2.59i)T+(−2.5+4.33i)T2 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2}
11 1+(−3+5.19i)T+(−5.5−9.52i)T2 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2}
13 1−T+13T2 1 - T + 13T^{2}
19 1+(−2−3.46i)T+(−9.5+16.4i)T2 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2}
23 1+(−2−3.46i)T+(−11.5+19.9i)T2 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2}
29 1−7T+29T2 1 - 7T + 29T^{2}
31 1+(3.5−6.06i)T+(−15.5−26.8i)T2 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2}
37 1+(4+6.92i)T+(−18.5+32.0i)T2 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2}
41 1+3T+41T2 1 + 3T + 41T^{2}
43 1+8T+43T2 1 + 8T + 43T^{2}
47 1+(−3.5−6.06i)T+(−23.5+40.7i)T2 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2}
53 1+(2−3.46i)T+(−26.5−45.8i)T2 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2}
59 1+(2.5−4.33i)T+(−29.5−51.0i)T2 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2}
61 1+(2+3.46i)T+(−30.5+52.8i)T2 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2}
67 1+(2−3.46i)T+(−33.5−58.0i)T2 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2}
71 1−16T+71T2 1 - 16T + 71T^{2}
73 1+(−1+1.73i)T+(−36.5−63.2i)T2 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2}
79 1+(−4−6.92i)T+(−39.5+68.4i)T2 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2}
83 1+9T+83T2 1 + 9T + 83T^{2}
89 1+(7+12.1i)T+(−44.5+77.0i)T2 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2}
97 1−8T+97T2 1 - 8T + 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

−10.12237919082952657319622433503, −9.257128849302038995483283601708, −8.393056621694735950159986182568, −7.11728869835905466669126672404, −6.47281252993319693230881983875, −5.98594335069302067964830398988, −5.33767484341052500650891695341, −3.65890322653304908679200282535, −2.90577556172763788980181151164, −1.49919855892003166702434593954, 1.15807201173852546747242214322, 2.22103824551955356977735315487, 3.48383452675271022971091152555, 4.47933589507849181220015900441, 5.00556484727790607526042897580, 6.59957024545222033655778502661, 6.99493873357664471222860795373, 8.165829775413167881415288830911, 9.141596268552397601724739856327, 9.767731668144289755730142869259

Graph of the ZZ-function along the critical line