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Properties

Label 2-189-7.2-c3-0-17
Degree 22
Conductor 189189
Sign 0.972−0.234i0.972 - 0.234i
Analytic cond. 11.151311.1513
Root an. cond. 3.339363.33936
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.38 + 4.12i)2-s + (−7.33 − 12.7i)4-s + (−9.35 + 16.1i)5-s + (14.2 − 11.8i)7-s + 31.8·8-s + (−44.5 − 77.1i)10-s + (−29.7 − 51.4i)11-s − 12.2·13-s + (15.1 + 86.8i)14-s + (−17.0 + 29.4i)16-s + (−5.76 − 9.97i)17-s + (−24.9 + 43.1i)19-s + 274.·20-s + 283.·22-s + (62.8 − 108. i)23-s + ⋯
L(s)  = 1  + (−0.841 + 1.45i)2-s + (−0.917 − 1.58i)4-s + (−0.836 + 1.44i)5-s + (0.767 − 0.641i)7-s + 1.40·8-s + (−1.40 − 2.43i)10-s + (−0.814 − 1.41i)11-s − 0.260·13-s + (0.289 + 1.65i)14-s + (−0.265 + 0.460i)16-s + (−0.0821 − 0.142i)17-s + (−0.300 + 0.520i)19-s + 3.06·20-s + 2.74·22-s + (0.570 − 0.987i)23-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((0.972−0.234i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+3/2) L(s)=((0.972−0.234i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: 0.972−0.234i0.972 - 0.234i
Analytic conductor: 11.151311.1513
Root analytic conductor: 3.339363.33936
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ189(163,⋅)\chi_{189} (163, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :3/2), 0.972−0.234i)(2,\ 189,\ (\ :3/2),\ 0.972 - 0.234i)

Particular Values

L(2)L(2) ≈\approx 0.535755+0.0635795i0.535755 + 0.0635795i
L(12)L(\frac12) ≈\approx 0.535755+0.0635795i0.535755 + 0.0635795i
L(52)L(\frac{5}{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+(−14.2+11.8i)T 1 + (-14.2 + 11.8i)T
good2 1+(2.38−4.12i)T+(−4−6.92i)T2 1 + (2.38 - 4.12i)T + (-4 - 6.92i)T^{2}
5 1+(9.35−16.1i)T+(−62.5−108.i)T2 1 + (9.35 - 16.1i)T + (-62.5 - 108. i)T^{2}
11 1+(29.7+51.4i)T+(−665.5+1.15e3i)T2 1 + (29.7 + 51.4i)T + (-665.5 + 1.15e3i)T^{2}
13 1+12.2T+2.19e3T2 1 + 12.2T + 2.19e3T^{2}
17 1+(5.76+9.97i)T+(−2.45e3+4.25e3i)T2 1 + (5.76 + 9.97i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(24.9−43.1i)T+(−3.42e3−5.94e3i)T2 1 + (24.9 - 43.1i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−62.8+108.i)T+(−6.08e3−1.05e4i)T2 1 + (-62.8 + 108. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1−228.T+2.43e4T2 1 - 228.T + 2.43e4T^{2}
31 1+(−94.7−164.i)T+(−1.48e4+2.57e4i)T2 1 + (-94.7 - 164. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(16.5−28.7i)T+(−2.53e4−4.38e4i)T2 1 + (16.5 - 28.7i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+524.T+6.89e4T2 1 + 524.T + 6.89e4T^{2}
43 1−234.T+7.95e4T2 1 - 234.T + 7.95e4T^{2}
47 1+(−136.+237.i)T+(−5.19e4−8.99e4i)T2 1 + (-136. + 237. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(127.+221.i)T+(−7.44e4+1.28e5i)T2 1 + (127. + 221. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(84.4+146.i)T+(−1.02e5+1.77e5i)T2 1 + (84.4 + 146. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−97.5+168.i)T+(−1.13e5−1.96e5i)T2 1 + (-97.5 + 168. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(257.+446.i)T+(−1.50e5+2.60e5i)T2 1 + (257. + 446. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−319.T+3.57e5T2 1 - 319.T + 3.57e5T^{2}
73 1+(317.+550.i)T+(−1.94e5+3.36e5i)T2 1 + (317. + 550. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(−426.+737.i)T+(−2.46e5−4.26e5i)T2 1 + (-426. + 737. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+264.T+5.71e5T2 1 + 264.T + 5.71e5T^{2}
89 1+(−457.+791.i)T+(−3.52e5−6.10e5i)T2 1 + (-457. + 791. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+455.T+9.12e5T2 1 + 455.T + 9.12e5T^{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

−11.85764169746453579793117541403, −10.66008509520163966763561025947, −10.37410328598890444273678540529, −8.507711338523717572048444223253, −8.044001704165547305186488762652, −7.07168240971046728719434960148, −6.34913660324455426958195865379, −4.90141405043495754226592101141, −3.16220010009531246388085761896, −0.36653202769270927681933868269, 1.19507446400328842937925994278, 2.47493151837956601806302794081, 4.29746553377812740709549112072, 5.07776213512682416163572887746, 7.60131774498690396179805214445, 8.374428640019347465456452249464, 9.132543974382755802662318424878, 10.07074679045479881539102014966, 11.24693320670175339103355874324, 12.05241024576072855203139251832

Graph of the ZZ-function along the critical line