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Properties

Label 2-161-7.2-c3-0-23
Degree 22
Conductor 161161
Sign 0.537−0.842i0.537 - 0.842i
Analytic cond. 9.499309.49930
Root an. cond. 3.082093.08209
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  + (0.674 − 1.16i)2-s + (3.89 + 6.74i)3-s + (3.08 + 5.35i)4-s + (3.28 − 5.68i)5-s + 10.5·6-s + (14.9 + 10.9i)7-s + 19.1·8-s + (−16.8 + 29.1i)9-s + (−4.43 − 7.67i)10-s + (−20.1 − 34.9i)11-s + (−24.0 + 41.6i)12-s + 33.8·13-s + (22.8 − 10.0i)14-s + 51.1·15-s + (−11.8 + 20.4i)16-s + (−48.0 − 83.3i)17-s + ⋯
L(s)  = 1  + (0.238 − 0.413i)2-s + (0.749 + 1.29i)3-s + (0.386 + 0.668i)4-s + (0.293 − 0.508i)5-s + 0.714·6-s + (0.807 + 0.590i)7-s + 0.845·8-s + (−0.622 + 1.07i)9-s + (−0.140 − 0.242i)10-s + (−0.553 − 0.958i)11-s + (−0.578 + 1.00i)12-s + 0.721·13-s + (0.436 − 0.192i)14-s + 0.880·15-s + (−0.184 + 0.319i)16-s + (−0.686 − 1.18i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((0.537−0.842i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.842i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+3/2) L(s)=((0.537−0.842i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.537 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: 0.537−0.842i0.537 - 0.842i
Analytic conductor: 9.499309.49930
Root analytic conductor: 3.082093.08209
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ161(93,⋅)\chi_{161} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :3/2), 0.537−0.842i)(2,\ 161,\ (\ :3/2),\ 0.537 - 0.842i)

Particular Values

L(2)L(2) ≈\approx 2.59334+1.42152i2.59334 + 1.42152i
L(12)L(\frac12) ≈\approx 2.59334+1.42152i2.59334 + 1.42152i
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)
bad7 1+(−14.9−10.9i)T 1 + (-14.9 - 10.9i)T
23 1+(−11.5+19.9i)T 1 + (-11.5 + 19.9i)T
good2 1+(−0.674+1.16i)T+(−4−6.92i)T2 1 + (-0.674 + 1.16i)T + (-4 - 6.92i)T^{2}
3 1+(−3.89−6.74i)T+(−13.5+23.3i)T2 1 + (-3.89 - 6.74i)T + (-13.5 + 23.3i)T^{2}
5 1+(−3.28+5.68i)T+(−62.5−108.i)T2 1 + (-3.28 + 5.68i)T + (-62.5 - 108. i)T^{2}
11 1+(20.1+34.9i)T+(−665.5+1.15e3i)T2 1 + (20.1 + 34.9i)T + (-665.5 + 1.15e3i)T^{2}
13 1−33.8T+2.19e3T2 1 - 33.8T + 2.19e3T^{2}
17 1+(48.0+83.3i)T+(−2.45e3+4.25e3i)T2 1 + (48.0 + 83.3i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(60.7−105.i)T+(−3.42e3−5.94e3i)T2 1 + (60.7 - 105. i)T + (-3.42e3 - 5.94e3i)T^{2}
29 1+173.T+2.43e4T2 1 + 173.T + 2.43e4T^{2}
31 1+(108.+188.i)T+(−1.48e4+2.57e4i)T2 1 + (108. + 188. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(−83.5+144.i)T+(−2.53e4−4.38e4i)T2 1 + (-83.5 + 144. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−12.2T+6.89e4T2 1 - 12.2T + 6.89e4T^{2}
43 1−463.T+7.95e4T2 1 - 463.T + 7.95e4T^{2}
47 1+(−25.9+45.0i)T+(−5.19e4−8.99e4i)T2 1 + (-25.9 + 45.0i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(340.+589.i)T+(−7.44e4+1.28e5i)T2 1 + (340. + 589. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(156.+270.i)T+(−1.02e5+1.77e5i)T2 1 + (156. + 270. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(176.−304.i)T+(−1.13e5−1.96e5i)T2 1 + (176. - 304. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(380.+659.i)T+(−1.50e5+2.60e5i)T2 1 + (380. + 659. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+231.T+3.57e5T2 1 + 231.T + 3.57e5T^{2}
73 1+(−354.−614.i)T+(−1.94e5+3.36e5i)T2 1 + (-354. - 614. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(−483.+837.i)T+(−2.46e5−4.26e5i)T2 1 + (-483. + 837. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+185.T+5.71e5T2 1 + 185.T + 5.71e5T^{2}
89 1+(684.−1.18e3i)T+(−3.52e5−6.10e5i)T2 1 + (684. - 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1−419.T+9.12e5T2 1 - 419.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

−12.62893827598609591816844085109, −11.23239648577201643621145573720, −10.86568849117051752539853675125, −9.369346277633209413094264285716, −8.632678308110454303923624323988, −7.78905709604838881107734367356, −5.69735471433146362475785506581, −4.49882315071961843510796469210, −3.44812805233385238507138618250, −2.18145381692967551969338183233, 1.43869400251047523563165211544, 2.37680812775363862851903572523, 4.54584784771482425131242366736, 6.15313956615447258019189440101, 7.02475044992204292254137600843, 7.69428781587778136693114740890, 8.880999709977507561166345519053, 10.53377557959153544032607472821, 11.04081172414880113948204978023, 12.68069785863097005388254797846

Graph of the ZZ-function along the critical line