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Properties

Label 2-161-161.100-c1-0-0
Degree 22
Conductor 161161
Sign −0.177−0.984i-0.177 - 0.984i
Analytic cond. 1.285591.28559
Root an. cond. 1.133831.13383
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  + (−1.50 − 1.18i)2-s + (−0.564 + 0.792i)3-s + (0.396 + 1.63i)4-s + (−0.812 + 0.156i)5-s + (1.79 − 0.526i)6-s + (−1.33 − 2.28i)7-s + (−0.253 + 0.554i)8-s + (0.671 + 1.94i)9-s + (1.41 + 0.727i)10-s + (−3.74 + 2.94i)11-s + (−1.52 − 0.608i)12-s + (−3.75 + 2.41i)13-s + (−0.688 + 5.03i)14-s + (0.334 − 0.732i)15-s + (4.03 − 2.07i)16-s + (4.06 + 3.87i)17-s + ⋯
L(s)  = 1  + (−1.06 − 0.838i)2-s + (−0.325 + 0.457i)3-s + (0.198 + 0.817i)4-s + (−0.363 + 0.0700i)5-s + (0.731 − 0.214i)6-s + (−0.505 − 0.862i)7-s + (−0.0894 + 0.195i)8-s + (0.223 + 0.646i)9-s + (0.446 + 0.230i)10-s + (−1.12 + 0.888i)11-s + (−0.438 − 0.175i)12-s + (−1.04 + 0.669i)13-s + (−0.184 + 1.34i)14-s + (0.0863 − 0.189i)15-s + (1.00 − 0.519i)16-s + (0.986 + 0.940i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((−0.177−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+1/2) L(s)=((−0.177−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.177−0.984i-0.177 - 0.984i
Analytic conductor: 1.285591.28559
Root analytic conductor: 1.133831.13383
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ161(100,⋅)\chi_{161} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :1/2), −0.177−0.984i)(2,\ 161,\ (\ :1/2),\ -0.177 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 0.134388+0.160774i0.134388 + 0.160774i
L(12)L(\frac12) ≈\approx 0.134388+0.160774i0.134388 + 0.160774i
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)
bad7 1+(1.33+2.28i)T 1 + (1.33 + 2.28i)T
23 1+(4.76+0.569i)T 1 + (4.76 + 0.569i)T
good2 1+(1.50+1.18i)T+(0.471+1.94i)T2 1 + (1.50 + 1.18i)T + (0.471 + 1.94i)T^{2}
3 1+(0.564−0.792i)T+(−0.981−2.83i)T2 1 + (0.564 - 0.792i)T + (-0.981 - 2.83i)T^{2}
5 1+(0.812−0.156i)T+(4.64−1.85i)T2 1 + (0.812 - 0.156i)T + (4.64 - 1.85i)T^{2}
11 1+(3.74−2.94i)T+(2.59−10.6i)T2 1 + (3.74 - 2.94i)T + (2.59 - 10.6i)T^{2}
13 1+(3.75−2.41i)T+(5.40−11.8i)T2 1 + (3.75 - 2.41i)T + (5.40 - 11.8i)T^{2}
17 1+(−4.06−3.87i)T+(0.808+16.9i)T2 1 + (-4.06 - 3.87i)T + (0.808 + 16.9i)T^{2}
19 1+(−0.735+0.701i)T+(0.904−18.9i)T2 1 + (-0.735 + 0.701i)T + (0.904 - 18.9i)T^{2}
29 1+(6.21−1.82i)T+(24.3−15.6i)T2 1 + (6.21 - 1.82i)T + (24.3 - 15.6i)T^{2}
31 1+(−9.48+0.905i)T+(30.4−5.86i)T2 1 + (-9.48 + 0.905i)T + (30.4 - 5.86i)T^{2}
37 1+(3.19+9.23i)T+(−29.0+22.8i)T2 1 + (3.19 + 9.23i)T + (-29.0 + 22.8i)T^{2}
41 1+(−1.65−1.91i)T+(−5.83+40.5i)T2 1 + (-1.65 - 1.91i)T + (-5.83 + 40.5i)T^{2}
43 1+(1.75+3.83i)T+(−28.1+32.4i)T2 1 + (1.75 + 3.83i)T + (-28.1 + 32.4i)T^{2}
47 1+(−1.59+2.76i)T+(−23.5−40.7i)T2 1 + (-1.59 + 2.76i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.0701+1.47i)T+(−52.7−5.03i)T2 1 + (-0.0701 + 1.47i)T + (-52.7 - 5.03i)T^{2}
59 1+(1.25+0.646i)T+(34.2+48.0i)T2 1 + (1.25 + 0.646i)T + (34.2 + 48.0i)T^{2}
61 1+(−3.59−5.04i)T+(−19.9+57.6i)T2 1 + (-3.59 - 5.04i)T + (-19.9 + 57.6i)T^{2}
67 1+(−0.641+0.256i)T+(48.4−46.2i)T2 1 + (-0.641 + 0.256i)T + (48.4 - 46.2i)T^{2}
71 1+(−0.382+2.65i)T+(−68.1−20.0i)T2 1 + (-0.382 + 2.65i)T + (-68.1 - 20.0i)T^{2}
73 1+(−3.48−14.3i)T+(−64.8+33.4i)T2 1 + (-3.48 - 14.3i)T + (-64.8 + 33.4i)T^{2}
79 1+(−0.428−8.99i)T+(−78.6+7.50i)T2 1 + (-0.428 - 8.99i)T + (-78.6 + 7.50i)T^{2}
83 1+(−0.874+1.00i)T+(−11.8−82.1i)T2 1 + (-0.874 + 1.00i)T + (-11.8 - 82.1i)T^{2}
89 1+(−9.14−0.872i)T+(87.3+16.8i)T2 1 + (-9.14 - 0.872i)T + (87.3 + 16.8i)T^{2}
97 1+(−2.97−3.43i)T+(−13.8+96.0i)T2 1 + (-2.97 - 3.43i)T + (-13.8 + 96.0i)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

−12.78465505982718584968929705735, −11.88398371175339884826873502445, −10.78815826103341232979571788747, −10.08604476205786934468093878813, −9.694011055127124032125863108122, −7.991181612172926085506579763720, −7.36091114798159858837211087006, −5.38920733770409123735736204866, −4.00387396687050347475728501831, −2.17556727431797961745889276513, 0.27691127081737558956578050787, 3.12463736017775270383203340264, 5.47728887601823494634803495798, 6.33898317807467125044176465754, 7.63273086371976159119890259690, 8.145610849529434503241163268017, 9.511486221509587833970891671678, 10.10016433123245819428626494249, 11.84391089271309400579178396065, 12.35961875432616687131711209135

Graph of the ZZ-function along the critical line