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Properties

Label 2-117-9.5-c2-0-16
Degree 22
Conductor 117117
Sign −0.629+0.776i-0.629 + 0.776i
Analytic cond. 3.188013.18801
Root an. cond. 1.785501.78550
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.22 + 1.86i)2-s + (0.496 − 2.95i)3-s + (4.93 − 8.54i)4-s + (0.733 + 0.423i)5-s + (3.90 + 10.4i)6-s + (−2.32 − 4.01i)7-s + 21.8i·8-s + (−8.50 − 2.93i)9-s − 3.15·10-s + (−10.6 + 6.16i)11-s + (−22.8 − 18.8i)12-s + (−1.80 + 3.12i)13-s + (14.9 + 8.63i)14-s + (1.61 − 1.96i)15-s + (−20.9 − 36.2i)16-s − 25.9i·17-s + ⋯
L(s)  = 1  + (−1.61 + 0.930i)2-s + (0.165 − 0.986i)3-s + (1.23 − 2.13i)4-s + (0.146 + 0.0847i)5-s + (0.651 + 1.74i)6-s + (−0.331 − 0.574i)7-s + 2.73i·8-s + (−0.945 − 0.326i)9-s − 0.315·10-s + (−0.970 + 0.560i)11-s + (−1.90 − 1.56i)12-s + (−0.138 + 0.240i)13-s + (1.06 + 0.617i)14-s + (0.107 − 0.130i)15-s + (−1.30 − 2.26i)16-s − 1.52i·17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((−0.629+0.776i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((−0.629+0.776i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: −0.629+0.776i-0.629 + 0.776i
Analytic conductor: 3.188013.18801
Root analytic conductor: 1.785501.78550
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ117(14,⋅)\chi_{117} (14, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1), −0.629+0.776i)(2,\ 117,\ (\ :1),\ -0.629 + 0.776i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.123980−0.260159i0.123980 - 0.260159i
L(12)L(\frac12) ≈\approx 0.123980−0.260159i0.123980 - 0.260159i
L(2)L(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+(−0.496+2.95i)T 1 + (-0.496 + 2.95i)T
13 1+(1.80−3.12i)T 1 + (1.80 - 3.12i)T
good2 1+(3.22−1.86i)T+(2−3.46i)T2 1 + (3.22 - 1.86i)T + (2 - 3.46i)T^{2}
5 1+(−0.733−0.423i)T+(12.5+21.6i)T2 1 + (-0.733 - 0.423i)T + (12.5 + 21.6i)T^{2}
7 1+(2.32+4.01i)T+(−24.5+42.4i)T2 1 + (2.32 + 4.01i)T + (-24.5 + 42.4i)T^{2}
11 1+(10.6−6.16i)T+(60.5−104.i)T2 1 + (10.6 - 6.16i)T + (60.5 - 104. i)T^{2}
17 1+25.9iT−289T2 1 + 25.9iT - 289T^{2}
19 1+12.4T+361T2 1 + 12.4T + 361T^{2}
23 1+(−0.958−0.553i)T+(264.5+458.i)T2 1 + (-0.958 - 0.553i)T + (264.5 + 458. i)T^{2}
29 1+(40.9−23.6i)T+(420.5−728.i)T2 1 + (40.9 - 23.6i)T + (420.5 - 728. i)T^{2}
31 1+(−5.53+9.57i)T+(−480.5−832.i)T2 1 + (-5.53 + 9.57i)T + (-480.5 - 832. i)T^{2}
37 1−4.57T+1.36e3T2 1 - 4.57T + 1.36e3T^{2}
41 1+(64.3+37.1i)T+(840.5+1.45e3i)T2 1 + (64.3 + 37.1i)T + (840.5 + 1.45e3i)T^{2}
43 1+(−35.0−60.6i)T+(−924.5+1.60e3i)T2 1 + (-35.0 - 60.6i)T + (-924.5 + 1.60e3i)T^{2}
47 1+(−61.9+35.7i)T+(1.10e3−1.91e3i)T2 1 + (-61.9 + 35.7i)T + (1.10e3 - 1.91e3i)T^{2}
53 1+88.9iT−2.80e3T2 1 + 88.9iT - 2.80e3T^{2}
59 1+(−35.0−20.2i)T+(1.74e3+3.01e3i)T2 1 + (-35.0 - 20.2i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(−29.6−51.4i)T+(−1.86e3+3.22e3i)T2 1 + (-29.6 - 51.4i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(−11.8+20.5i)T+(−2.24e3−3.88e3i)T2 1 + (-11.8 + 20.5i)T + (-2.24e3 - 3.88e3i)T^{2}
71 1−49.7iT−5.04e3T2 1 - 49.7iT - 5.04e3T^{2}
73 1−10.8T+5.32e3T2 1 - 10.8T + 5.32e3T^{2}
79 1+(3.54+6.13i)T+(−3.12e3+5.40e3i)T2 1 + (3.54 + 6.13i)T + (-3.12e3 + 5.40e3i)T^{2}
83 1+(−14.0+8.12i)T+(3.44e3−5.96e3i)T2 1 + (-14.0 + 8.12i)T + (3.44e3 - 5.96e3i)T^{2}
89 1+31.1iT−7.92e3T2 1 + 31.1iT - 7.92e3T^{2}
97 1+(50.9+88.3i)T+(−4.70e3+8.14e3i)T2 1 + (50.9 + 88.3i)T + (-4.70e3 + 8.14e3i)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

−13.09272702501050001521765423542, −11.61537163505527065814025259197, −10.43171466232481415471789801602, −9.501482206345124253927602924283, −8.406308601124964929815843634085, −7.34662348328921715678831028908, −6.87966183936390884572052520286, −5.51099735199299567248936870723, −2.21252598026336339194040896378, −0.31051741515279282621295833119, 2.35031573048095493662594938460, 3.65349316292417534809230668172, 5.81354095803711066459364550684, 7.83535571128855652578464164568, 8.664873066810216427468746708056, 9.496834204690298509979137801990, 10.47751103236856546901472420569, 11.00861079911640623116216775424, 12.24904345565588201305269633195, 13.29204858667899789211492765726

Graph of the ZZ-function along the critical line