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Properties

Label 2-882-7.4-c1-0-12
Degree 22
Conductor 882882
Sign −0.386+0.922i-0.386 + 0.922i
Analytic cond. 7.042807.04280
Root an. cond. 2.653822.65382
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 + (−0.499 + 0.866i)4-s + 0.999·8-s − 4·13-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (−1 − 1.73i)19-s + (2.5 − 4.33i)25-s + (2 + 3.46i)26-s + 6·29-s + (2 − 3.46i)31-s + (−0.499 + 0.866i)32-s − 6·34-s + (−1 − 1.73i)37-s + (−0.999 + 1.73i)38-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 0.353·8-s − 1.10·13-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.229 − 0.397i)19-s + (0.5 − 0.866i)25-s + (0.392 + 0.679i)26-s + 1.11·29-s + (0.359 − 0.622i)31-s + (−0.0883 + 0.153i)32-s − 1.02·34-s + (−0.164 − 0.284i)37-s + (−0.162 + 0.280i)38-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((−0.386+0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((−0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{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: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: −0.386+0.922i-0.386 + 0.922i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(361,⋅)\chi_{882} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), −0.386+0.922i)(2,\ 882,\ (\ :1/2),\ -0.386 + 0.922i)

Particular Values

L(1)L(1) ≈\approx 0.554939−0.834266i0.554939 - 0.834266i
L(12)L(\frac12) ≈\approx 0.554939−0.834266i0.554939 - 0.834266i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1 1
7 1 1
good5 1+(−2.5+4.33i)T2 1 + (-2.5 + 4.33i)T^{2}
11 1+(−5.5−9.52i)T2 1 + (-5.5 - 9.52i)T^{2}
13 1+4T+13T2 1 + 4T + 13T^{2}
17 1+(−3+5.19i)T+(−8.5−14.7i)T2 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2}
19 1+(1+1.73i)T+(−9.5+16.4i)T2 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1−6T+29T2 1 - 6T + 29T^{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+1.73i)T+(−18.5+32.0i)T2 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−8T+43T2 1 - 8T + 43T^{2}
47 1+(6+10.3i)T+(−23.5+40.7i)T2 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2}
53 1+(−3+5.19i)T+(−26.5−45.8i)T2 1 + (-3 + 5.19i)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+(4+6.92i)T+(−30.5+52.8i)T2 1 + (4 + 6.92i)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+71T2 1 + 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 1−6T+83T2 1 - 6T + 83T^{2}
89 1+(3+5.19i)T+(−44.5+77.0i)T2 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2}
97 1+10T+97T2 1 + 10T + 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

−9.947962399167162574763516562225, −9.202208327398030096372454610694, −8.290591895518752341408238230342, −7.43214040408875706848220213907, −6.61190330864623345980003208225, −5.22717740194178482115636502177, −4.48973455666388034502967389192, −3.13792778408072637693602168647, −2.27167569372610041124362855296, −0.57480985419816511141810679089, 1.41601001243632920313198798025, 2.96914043439670374576015188211, 4.30707667531637213604764354311, 5.25382606869139825437544321754, 6.16705628437093657687964839083, 7.04484775800697037631545065049, 7.893845413274031277357094778620, 8.579070835181114491590579675972, 9.562839283642452873150273824419, 10.24050168938634619756520511423

Graph of the ZZ-function along the critical line