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Properties

Label 2-882-7.2-c1-0-6
Degree 22
Conductor 882882
Sign 0.968−0.250i0.968 - 0.250i
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)=((0.968−0.250i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((0.968−0.250i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 0.968−0.250i0.968 - 0.250i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(667,⋅)\chi_{882} (667, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), 0.968−0.250i)(2,\ 882,\ (\ :1/2),\ 0.968 - 0.250i)

Particular Values

L(1)L(1) ≈\approx 1.28668+0.163967i1.28668 + 0.163967i
L(12)L(\frac12) ≈\approx 1.28668+0.163967i1.28668 + 0.163967i
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

−10.03458217196504737449000820073, −9.077121502901718740174244911167, −8.657303558006121919107339150908, −7.50832315445137166254741236384, −6.86895803606696537734860737541, −5.91763966675191839057885172732, −5.01874547052489253303943963197, −3.96201716102204789433231923865, −2.58645387986878904012478155109, −0.905061614027266334376927730819, 1.13998151026283401648955805326, 2.44197013065682108198827134917, 3.67099487377760487735754976729, 4.47193653024812030918892525950, 5.83842875049177632595080084384, 6.63343478235961428792921448827, 7.82916649232246112174780567500, 8.584248611012648524612602635586, 9.174995102540854900453790019561, 10.39247262359629443990942940583

Graph of the ZZ-function along the critical line