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Properties

Label 2-153-153.113-c1-0-12
Degree 22
Conductor 153153
Sign −0.253+0.967i-0.253 + 0.967i
Analytic cond. 1.221711.22171
Root an. cond. 1.105311.10531
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  + (−0.801 + 0.105i)2-s + (1.57 − 0.719i)3-s + (−1.30 + 0.348i)4-s + (−1.37 − 2.79i)5-s + (−1.18 + 0.742i)6-s + (−4.52 − 2.23i)7-s + (2.49 − 1.03i)8-s + (1.96 − 2.26i)9-s + (1.39 + 2.09i)10-s + (0.780 − 0.265i)11-s + (−1.79 + 1.48i)12-s + (1.16 + 4.35i)13-s + (3.86 + 1.31i)14-s + (−4.17 − 3.40i)15-s + (0.439 − 0.253i)16-s + (3.78 − 1.62i)17-s + ⋯
L(s)  = 1  + (−0.566 + 0.0745i)2-s + (0.909 − 0.415i)3-s + (−0.650 + 0.174i)4-s + (−0.615 − 1.24i)5-s + (−0.484 + 0.303i)6-s + (−1.71 − 0.843i)7-s + (0.883 − 0.365i)8-s + (0.655 − 0.755i)9-s + (0.442 + 0.661i)10-s + (0.235 − 0.0799i)11-s + (−0.519 + 0.428i)12-s + (0.323 + 1.20i)13-s + (1.03 + 0.350i)14-s + (−1.07 − 0.880i)15-s + (0.109 − 0.0634i)16-s + (0.918 − 0.394i)17-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=((−0.253+0.967i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+1/2) L(s)=((−0.253+0.967i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: −0.253+0.967i-0.253 + 0.967i
Analytic conductor: 1.221711.22171
Root analytic conductor: 1.105311.10531
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ153(113,⋅)\chi_{153} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 153, ( :1/2), −0.253+0.967i)(2,\ 153,\ (\ :1/2),\ -0.253 + 0.967i)

Particular Values

L(1)L(1) ≈\approx 0.433494−0.561774i0.433494 - 0.561774i
L(12)L(\frac12) ≈\approx 0.433494−0.561774i0.433494 - 0.561774i
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+(−1.57+0.719i)T 1 + (-1.57 + 0.719i)T
17 1+(−3.78+1.62i)T 1 + (-3.78 + 1.62i)T
good2 1+(0.801−0.105i)T+(1.93−0.517i)T2 1 + (0.801 - 0.105i)T + (1.93 - 0.517i)T^{2}
5 1+(1.37+2.79i)T+(−3.04+3.96i)T2 1 + (1.37 + 2.79i)T + (-3.04 + 3.96i)T^{2}
7 1+(4.52+2.23i)T+(4.26+5.55i)T2 1 + (4.52 + 2.23i)T + (4.26 + 5.55i)T^{2}
11 1+(−0.780+0.265i)T+(8.72−6.69i)T2 1 + (-0.780 + 0.265i)T + (8.72 - 6.69i)T^{2}
13 1+(−1.16−4.35i)T+(−11.2+6.5i)T2 1 + (-1.16 - 4.35i)T + (-11.2 + 6.5i)T^{2}
19 1+(1.95+0.811i)T+(13.4+13.4i)T2 1 + (1.95 + 0.811i)T + (13.4 + 13.4i)T^{2}
23 1+(−2.61+2.98i)T+(−3.00−22.8i)T2 1 + (-2.61 + 2.98i)T + (-3.00 - 22.8i)T^{2}
29 1+(0.0274−0.418i)T+(−28.7−3.78i)T2 1 + (0.0274 - 0.418i)T + (-28.7 - 3.78i)T^{2}
31 1+(−0.426+1.25i)T+(−24.5−18.8i)T2 1 + (-0.426 + 1.25i)T + (-24.5 - 18.8i)T^{2}
37 1+(−0.0132−0.0664i)T+(−34.1+14.1i)T2 1 + (-0.0132 - 0.0664i)T + (-34.1 + 14.1i)T^{2}
41 1+(3.37−0.221i)T+(40.6−5.35i)T2 1 + (3.37 - 0.221i)T + (40.6 - 5.35i)T^{2}
43 1+(−5.74−4.40i)T+(11.1+41.5i)T2 1 + (-5.74 - 4.40i)T + (11.1 + 41.5i)T^{2}
47 1+(−1.39+5.19i)T+(−40.7−23.5i)T2 1 + (-1.39 + 5.19i)T + (-40.7 - 23.5i)T^{2}
53 1+(−2.70+6.52i)T+(−37.4−37.4i)T2 1 + (-2.70 + 6.52i)T + (-37.4 - 37.4i)T^{2}
59 1+(0.445−3.38i)T+(−56.9−15.2i)T2 1 + (0.445 - 3.38i)T + (-56.9 - 15.2i)T^{2}
61 1+(−0.762+1.54i)T+(−37.1−48.3i)T2 1 + (-0.762 + 1.54i)T + (-37.1 - 48.3i)T^{2}
67 1+(1.90+1.10i)T+(33.5+58.0i)T2 1 + (1.90 + 1.10i)T + (33.5 + 58.0i)T^{2}
71 1+(2.14−0.426i)T+(65.5−27.1i)T2 1 + (2.14 - 0.426i)T + (65.5 - 27.1i)T^{2}
73 1+(−4.79−3.20i)T+(27.9+67.4i)T2 1 + (-4.79 - 3.20i)T + (27.9 + 67.4i)T^{2}
79 1+(4.25+12.5i)T+(−62.6+48.0i)T2 1 + (4.25 + 12.5i)T + (-62.6 + 48.0i)T^{2}
83 1+(0.479+3.64i)T+(−80.1+21.4i)T2 1 + (0.479 + 3.64i)T + (-80.1 + 21.4i)T^{2}
89 1+(3.26−3.26i)T−89iT2 1 + (3.26 - 3.26i)T - 89iT^{2}
97 1+(−17.3−1.13i)T+(96.1+12.6i)T2 1 + (-17.3 - 1.13i)T + (96.1 + 12.6i)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

−12.93350414700559267097046782507, −12.07502844727960414080991217358, −10.14954012011888537517577418568, −9.251663720603465019051831464120, −8.778367747398176203087437004402, −7.63192594678361177680063111492, −6.66360676366364220867028988226, −4.41312452235218146017256165881, −3.56596897233861041543756300222, −0.809196702738645643404852072570, 2.93884789598199690760220663709, 3.72629409301597353209936890430, 5.74635690537699758380372905212, 7.21796951779256621287709375883, 8.276194188956415522531295274280, 9.278533092660252049869840102174, 10.08995055018183252078846437065, 10.73643953059534397890872881791, 12.45990676187703832805807467431, 13.30508488415132205311310052057

Graph of the ZZ-function along the critical line