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Properties

Label 2-117-117.86-c1-0-2
Degree 22
Conductor 117117
Sign 0.922+0.385i0.922 + 0.385i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
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  + (−2.24 − 0.600i)2-s + (0.724 − 1.57i)3-s + (2.92 + 1.69i)4-s + (1.01 + 3.78i)5-s + (−2.56 + 3.09i)6-s + (2.27 + 0.608i)7-s + (−2.26 − 2.26i)8-s + (−1.95 − 2.27i)9-s − 9.08i·10-s + (0.637 − 2.38i)11-s + (4.78 − 3.38i)12-s + (1.99 + 3.00i)13-s + (−4.72 − 2.72i)14-s + (6.68 + 1.14i)15-s + (0.335 + 0.581i)16-s + 0.901·17-s + ⋯
L(s)  = 1  + (−1.58 − 0.424i)2-s + (0.418 − 0.908i)3-s + (1.46 + 0.845i)4-s + (0.453 + 1.69i)5-s + (−1.04 + 1.26i)6-s + (0.858 + 0.230i)7-s + (−0.801 − 0.801i)8-s + (−0.650 − 0.759i)9-s − 2.87i·10-s + (0.192 − 0.717i)11-s + (1.38 − 0.976i)12-s + (0.553 + 0.832i)13-s + (−1.26 − 0.729i)14-s + (1.72 + 0.295i)15-s + (0.0838 + 0.145i)16-s + 0.218·17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.922+0.385i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.922+0.385i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.922+0.385i0.922 + 0.385i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(86,⋅)\chi_{117} (86, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.922+0.385i)(2,\ 117,\ (\ :1/2),\ 0.922 + 0.385i)

Particular Values

L(1)L(1) ≈\approx 0.656818−0.131723i0.656818 - 0.131723i
L(12)L(\frac12) ≈\approx 0.656818−0.131723i0.656818 - 0.131723i
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+(−0.724+1.57i)T 1 + (-0.724 + 1.57i)T
13 1+(−1.99−3.00i)T 1 + (-1.99 - 3.00i)T
good2 1+(2.24+0.600i)T+(1.73+i)T2 1 + (2.24 + 0.600i)T + (1.73 + i)T^{2}
5 1+(−1.01−3.78i)T+(−4.33+2.5i)T2 1 + (-1.01 - 3.78i)T + (-4.33 + 2.5i)T^{2}
7 1+(−2.27−0.608i)T+(6.06+3.5i)T2 1 + (-2.27 - 0.608i)T + (6.06 + 3.5i)T^{2}
11 1+(−0.637+2.38i)T+(−9.52−5.5i)T2 1 + (-0.637 + 2.38i)T + (-9.52 - 5.5i)T^{2}
17 1−0.901T+17T2 1 - 0.901T + 17T^{2}
19 1+(2.07+2.07i)T+19iT2 1 + (2.07 + 2.07i)T + 19iT^{2}
23 1+(−1.50+2.60i)T+(−11.5−19.9i)T2 1 + (-1.50 + 2.60i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.19+1.26i)T+(14.5−25.1i)T2 1 + (-2.19 + 1.26i)T + (14.5 - 25.1i)T^{2}
31 1+(−1.92+0.516i)T+(26.8−15.5i)T2 1 + (-1.92 + 0.516i)T + (26.8 - 15.5i)T^{2}
37 1+(7.88−7.88i)T−37iT2 1 + (7.88 - 7.88i)T - 37iT^{2}
41 1+(0.895+3.34i)T+(−35.5+20.5i)T2 1 + (0.895 + 3.34i)T + (-35.5 + 20.5i)T^{2}
43 1+(−5.11+2.95i)T+(21.5−37.2i)T2 1 + (-5.11 + 2.95i)T + (21.5 - 37.2i)T^{2}
47 1+(−0.259+0.966i)T+(−40.7−23.5i)T2 1 + (-0.259 + 0.966i)T + (-40.7 - 23.5i)T^{2}
53 1+0.635iT−53T2 1 + 0.635iT - 53T^{2}
59 1+(5.54−1.48i)T+(51.0−29.5i)T2 1 + (5.54 - 1.48i)T + (51.0 - 29.5i)T^{2}
61 1+(4.38+7.58i)T+(−30.5+52.8i)T2 1 + (4.38 + 7.58i)T + (-30.5 + 52.8i)T^{2}
67 1+(8.84−2.37i)T+(58.0−33.5i)T2 1 + (8.84 - 2.37i)T + (58.0 - 33.5i)T^{2}
71 1+(7.93−7.93i)T−71iT2 1 + (7.93 - 7.93i)T - 71iT^{2}
73 1+(−9.16+9.16i)T−73iT2 1 + (-9.16 + 9.16i)T - 73iT^{2}
79 1+(0.204+0.353i)T+(−39.5+68.4i)T2 1 + (0.204 + 0.353i)T + (-39.5 + 68.4i)T^{2}
83 1+(3.66+0.982i)T+(71.8+41.5i)T2 1 + (3.66 + 0.982i)T + (71.8 + 41.5i)T^{2}
89 1+(10.0+10.0i)T+89iT2 1 + (10.0 + 10.0i)T + 89iT^{2}
97 1+(0.733−2.73i)T+(−84.0−48.5i)T2 1 + (0.733 - 2.73i)T + (-84.0 - 48.5i)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

−13.73023864220123120683254535369, −11.86556846579138698335888656222, −11.18966689120698934304052080258, −10.39647051415069651888665537103, −9.048305871941290891006053587852, −8.240945338428186487694289612719, −7.11260518813545481899740024845, −6.32849185226057388628628728474, −2.97951775036803070516497326471, −1.81165844178678455103924968030, 1.52509719836628135297247262926, 4.45964904925987665761220357103, 5.65054115754548528664990125883, 7.72700055771208149616292663263, 8.479533100890136051194788873705, 9.169177496443813941043255271024, 10.05001158092943499196948095233, 10.96529773834811789960097845640, 12.46950227425898288849391338489, 13.76522553668930932826193898755

Graph of the ZZ-function along the critical line