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Properties

Label 2-171-171.110-c1-0-14
Degree 22
Conductor 171171
Sign 0.670+0.742i0.670 + 0.742i
Analytic cond. 1.365441.36544
Root an. cond. 1.168521.16852
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  + (1.98 − 1.66i)2-s + (0.915 + 1.47i)3-s + (0.819 − 4.64i)4-s + (−1.06 + 2.92i)5-s + (4.26 + 1.39i)6-s + (−1.52 − 2.64i)7-s + (−3.52 − 6.09i)8-s + (−1.32 + 2.69i)9-s + (2.76 + 7.58i)10-s − 3.42i·11-s + (7.57 − 3.05i)12-s + (0.864 + 2.37i)13-s + (−7.42 − 2.70i)14-s + (−5.27 + 1.11i)15-s + (−8.28 − 3.01i)16-s + (−1.79 + 4.92i)17-s + ⋯
L(s)  = 1  + (1.40 − 1.17i)2-s + (0.528 + 0.848i)3-s + (0.409 − 2.32i)4-s + (−0.476 + 1.30i)5-s + (1.74 + 0.568i)6-s + (−0.576 − 0.998i)7-s + (−1.24 − 2.15i)8-s + (−0.440 + 0.897i)9-s + (0.872 + 2.39i)10-s − 1.03i·11-s + (2.18 − 0.880i)12-s + (0.239 + 0.658i)13-s + (−1.98 − 0.722i)14-s + (−1.36 + 0.287i)15-s + (−2.07 − 0.754i)16-s + (−0.434 + 1.19i)17-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=((0.670+0.742i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+1/2) L(s)=((0.670+0.742i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: 0.670+0.742i0.670 + 0.742i
Analytic conductor: 1.365441.36544
Root analytic conductor: 1.168521.16852
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ171(110,⋅)\chi_{171} (110, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 171, ( :1/2), 0.670+0.742i)(2,\ 171,\ (\ :1/2),\ 0.670 + 0.742i)

Particular Values

L(1)L(1) ≈\approx 2.08961−0.928351i2.08961 - 0.928351i
L(12)L(\frac12) ≈\approx 2.08961−0.928351i2.08961 - 0.928351i
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)
bad3 1+(−0.915−1.47i)T 1 + (-0.915 - 1.47i)T
19 1+(3.57+2.49i)T 1 + (3.57 + 2.49i)T
good2 1+(−1.98+1.66i)T+(0.347−1.96i)T2 1 + (-1.98 + 1.66i)T + (0.347 - 1.96i)T^{2}
5 1+(1.06−2.92i)T+(−3.83−3.21i)T2 1 + (1.06 - 2.92i)T + (-3.83 - 3.21i)T^{2}
7 1+(1.52+2.64i)T+(−3.5+6.06i)T2 1 + (1.52 + 2.64i)T + (-3.5 + 6.06i)T^{2}
11 1+3.42iT−11T2 1 + 3.42iT - 11T^{2}
13 1+(−0.864−2.37i)T+(−9.95+8.35i)T2 1 + (-0.864 - 2.37i)T + (-9.95 + 8.35i)T^{2}
17 1+(1.79−4.92i)T+(−13.0−10.9i)T2 1 + (1.79 - 4.92i)T + (-13.0 - 10.9i)T^{2}
23 1+(−2.96−0.522i)T+(21.6+7.86i)T2 1 + (-2.96 - 0.522i)T + (21.6 + 7.86i)T^{2}
29 1+(0.536−3.04i)T+(−27.2−9.91i)T2 1 + (0.536 - 3.04i)T + (-27.2 - 9.91i)T^{2}
31 1+0.794iT−31T2 1 + 0.794iT - 31T^{2}
37 1+7.02iT−37T2 1 + 7.02iT - 37T^{2}
41 1+(−4.12+3.46i)T+(7.11−40.3i)T2 1 + (-4.12 + 3.46i)T + (7.11 - 40.3i)T^{2}
43 1+(0.331+1.88i)T+(−40.4+14.7i)T2 1 + (0.331 + 1.88i)T + (-40.4 + 14.7i)T^{2}
47 1+(−12.8−2.26i)T+(44.1+16.0i)T2 1 + (-12.8 - 2.26i)T + (44.1 + 16.0i)T^{2}
53 1+(−3.01−2.53i)T+(9.20+52.1i)T2 1 + (-3.01 - 2.53i)T + (9.20 + 52.1i)T^{2}
59 1+(0.194+1.10i)T+(−55.4+20.1i)T2 1 + (0.194 + 1.10i)T + (-55.4 + 20.1i)T^{2}
61 1+(1.80−0.655i)T+(46.7−39.2i)T2 1 + (1.80 - 0.655i)T + (46.7 - 39.2i)T^{2}
67 1+(0.412−0.491i)T+(−11.6−65.9i)T2 1 + (0.412 - 0.491i)T + (-11.6 - 65.9i)T^{2}
71 1+(5.62−4.72i)T+(12.3−69.9i)T2 1 + (5.62 - 4.72i)T + (12.3 - 69.9i)T^{2}
73 1+(−0.808−4.58i)T+(−68.5+24.9i)T2 1 + (-0.808 - 4.58i)T + (-68.5 + 24.9i)T^{2}
79 1+(2.33−6.41i)T+(−60.5−50.7i)T2 1 + (2.33 - 6.41i)T + (-60.5 - 50.7i)T^{2}
83 1+(1.59−0.918i)T+(41.5−71.8i)T2 1 + (1.59 - 0.918i)T + (41.5 - 71.8i)T^{2}
89 1+(−2.03+11.5i)T+(−83.6−30.4i)T2 1 + (-2.03 + 11.5i)T + (-83.6 - 30.4i)T^{2}
97 1+(9.52+11.3i)T+(−16.8+95.5i)T2 1 + (9.52 + 11.3i)T + (-16.8 + 95.5i)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

−12.82198994229207188970906133883, −11.26557216166821284544930107685, −10.81843098249841656447365449014, −10.36178907271571852825500870562, −8.933921227648752327218134862112, −7.05865010962978537891688211138, −5.90620644586371051832012080784, −4.14104580933507858536697868680, −3.69063208915605254230255907950, −2.61315046738754447422858556097, 2.74769515233384640201927508964, 4.29519186936832943303082880676, 5.39051103503652240538669954350, 6.45522344545010131963087835525, 7.53538850035874915860778105243, 8.416519691851872697336088383125, 9.245841628767532322224238918102, 11.89388735658571010887104156842, 12.38060476896402524028946297027, 12.95898318351921147727940315993

Graph of the ZZ-function along the critical line