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Properties

Label 2-161-161.100-c1-0-13
Degree 22
Conductor 161161
Sign −0.952−0.305i-0.952 - 0.305i
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  + (−1.59 − 1.25i)2-s + (1.76 − 2.47i)3-s + (0.502 + 2.07i)4-s + (−2.84 + 0.549i)5-s + (−5.92 + 1.73i)6-s + (−2.63 + 0.260i)7-s + (0.110 − 0.242i)8-s + (−2.03 − 5.88i)9-s + (5.24 + 2.70i)10-s + (0.552 − 0.434i)11-s + (6.00 + 2.40i)12-s + (3.04 − 1.95i)13-s + (4.53 + 2.89i)14-s + (−3.66 + 8.01i)15-s + (3.30 − 1.70i)16-s + (−3.27 − 3.11i)17-s + ⋯
L(s)  = 1  + (−1.12 − 0.888i)2-s + (1.01 − 1.42i)3-s + (0.251 + 1.03i)4-s + (−1.27 + 0.245i)5-s + (−2.41 + 0.709i)6-s + (−0.995 + 0.0982i)7-s + (0.0390 − 0.0855i)8-s + (−0.678 − 1.96i)9-s + (1.65 + 0.854i)10-s + (0.166 − 0.131i)11-s + (1.73 + 0.694i)12-s + (0.845 − 0.543i)13-s + (1.21 + 0.773i)14-s + (−0.945 + 2.06i)15-s + (0.826 − 0.426i)16-s + (−0.793 − 0.756i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((−0.952−0.305i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+1/2) L(s)=((−0.952−0.305i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.952−0.305i-0.952 - 0.305i
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), −0.952−0.305i)(2,\ 161,\ (\ :1/2),\ -0.952 - 0.305i)

Particular Values

L(1)L(1) ≈\approx 0.0834494+0.533426i0.0834494 + 0.533426i
L(12)L(\frac12) ≈\approx 0.0834494+0.533426i0.0834494 + 0.533426i
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+(2.63−0.260i)T 1 + (2.63 - 0.260i)T
23 1+(4.72+0.838i)T 1 + (4.72 + 0.838i)T
good2 1+(1.59+1.25i)T+(0.471+1.94i)T2 1 + (1.59 + 1.25i)T + (0.471 + 1.94i)T^{2}
3 1+(−1.76+2.47i)T+(−0.981−2.83i)T2 1 + (-1.76 + 2.47i)T + (-0.981 - 2.83i)T^{2}
5 1+(2.84−0.549i)T+(4.64−1.85i)T2 1 + (2.84 - 0.549i)T + (4.64 - 1.85i)T^{2}
11 1+(−0.552+0.434i)T+(2.59−10.6i)T2 1 + (-0.552 + 0.434i)T + (2.59 - 10.6i)T^{2}
13 1+(−3.04+1.95i)T+(5.40−11.8i)T2 1 + (-3.04 + 1.95i)T + (5.40 - 11.8i)T^{2}
17 1+(3.27+3.11i)T+(0.808+16.9i)T2 1 + (3.27 + 3.11i)T + (0.808 + 16.9i)T^{2}
19 1+(−2.90+2.76i)T+(0.904−18.9i)T2 1 + (-2.90 + 2.76i)T + (0.904 - 18.9i)T^{2}
29 1+(−8.16+2.39i)T+(24.3−15.6i)T2 1 + (-8.16 + 2.39i)T + (24.3 - 15.6i)T^{2}
31 1+(0.658−0.0628i)T+(30.4−5.86i)T2 1 + (0.658 - 0.0628i)T + (30.4 - 5.86i)T^{2}
37 1+(−2.87−8.31i)T+(−29.0+22.8i)T2 1 + (-2.87 - 8.31i)T + (-29.0 + 22.8i)T^{2}
41 1+(0.195+0.225i)T+(−5.83+40.5i)T2 1 + (0.195 + 0.225i)T + (-5.83 + 40.5i)T^{2}
43 1+(2.93+6.42i)T+(−28.1+32.4i)T2 1 + (2.93 + 6.42i)T + (-28.1 + 32.4i)T^{2}
47 1+(−3.07+5.31i)T+(−23.5−40.7i)T2 1 + (-3.07 + 5.31i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.00171−0.0359i)T+(−52.7−5.03i)T2 1 + (0.00171 - 0.0359i)T + (-52.7 - 5.03i)T^{2}
59 1+(−1.03−0.532i)T+(34.2+48.0i)T2 1 + (-1.03 - 0.532i)T + (34.2 + 48.0i)T^{2}
61 1+(0.748+1.05i)T+(−19.9+57.6i)T2 1 + (0.748 + 1.05i)T + (-19.9 + 57.6i)T^{2}
67 1+(1.11−0.447i)T+(48.4−46.2i)T2 1 + (1.11 - 0.447i)T + (48.4 - 46.2i)T^{2}
71 1+(1.99−13.8i)T+(−68.1−20.0i)T2 1 + (1.99 - 13.8i)T + (-68.1 - 20.0i)T^{2}
73 1+(0.289+1.19i)T+(−64.8+33.4i)T2 1 + (0.289 + 1.19i)T + (-64.8 + 33.4i)T^{2}
79 1+(0.169+3.54i)T+(−78.6+7.50i)T2 1 + (0.169 + 3.54i)T + (-78.6 + 7.50i)T^{2}
83 1+(−7.99+9.22i)T+(−11.8−82.1i)T2 1 + (-7.99 + 9.22i)T + (-11.8 - 82.1i)T^{2}
89 1+(−11.7−1.12i)T+(87.3+16.8i)T2 1 + (-11.7 - 1.12i)T + (87.3 + 16.8i)T^{2}
97 1+(1.51+1.74i)T+(−13.8+96.0i)T2 1 + (1.51 + 1.74i)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

−11.99832765774048033737459753180, −11.62889296154104776180716471263, −10.20902386788805182619384910863, −8.978602337722509480428068839138, −8.347618493089077265090184576842, −7.48658444997203497737744830024, −6.48195113586280397339360341800, −3.47459431210530590428012275045, −2.60162290782447156972204715984, −0.66506896995692757795492370831, 3.49645953302534030438001838024, 4.22225216576648467323937488592, 6.26085255965382024077056859692, 7.66540262472126346099398329415, 8.452041278237618130837779580678, 9.140982862167710014855839383747, 9.961993372211636215069096810778, 10.90334182371402359085923405590, 12.37240418504198635798309019092, 13.82017111697639109846352825367

Graph of the ZZ-function along the critical line