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Properties

Label 2-153-153.11-c1-0-8
Degree 22
Conductor 153153
Sign −0.154−0.988i-0.154 - 0.988i
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  + (1.34 + 1.74i)2-s + (1.37 + 1.05i)3-s + (−0.738 + 2.75i)4-s + (−1.77 − 0.116i)5-s + (0.0106 + 3.81i)6-s + (−0.295 − 4.50i)7-s + (−1.73 + 0.718i)8-s + (0.792 + 2.89i)9-s + (−2.17 − 3.25i)10-s + (−3.64 − 3.19i)11-s + (−3.91 + 3.01i)12-s + (3.24 + 0.868i)13-s + (7.47 − 6.55i)14-s + (−2.31 − 2.02i)15-s + (1.36 + 0.785i)16-s + (−1.37 + 3.88i)17-s + ⋯
L(s)  = 1  + (0.948 + 1.23i)2-s + (0.795 + 0.606i)3-s + (−0.369 + 1.37i)4-s + (−0.792 − 0.0519i)5-s + (0.00433 + 1.55i)6-s + (−0.111 − 1.70i)7-s + (−0.612 + 0.253i)8-s + (0.264 + 0.964i)9-s + (−0.686 − 1.02i)10-s + (−1.09 − 0.962i)11-s + (−1.12 + 0.871i)12-s + (0.899 + 0.240i)13-s + (1.99 − 1.75i)14-s + (−0.598 − 0.521i)15-s + (0.340 + 0.196i)16-s + (−0.332 + 0.942i)17-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=((−0.154−0.988i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+1/2) L(s)=((−0.154−0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: −0.154−0.988i-0.154 - 0.988i
Analytic conductor: 1.221711.22171
Root analytic conductor: 1.105311.10531
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ153(11,⋅)\chi_{153} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 153, ( :1/2), −0.154−0.988i)(2,\ 153,\ (\ :1/2),\ -0.154 - 0.988i)

Particular Values

L(1)L(1) ≈\approx 1.23798+1.44649i1.23798 + 1.44649i
L(12)L(\frac12) ≈\approx 1.23798+1.44649i1.23798 + 1.44649i
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+(−1.37−1.05i)T 1 + (-1.37 - 1.05i)T
17 1+(1.37−3.88i)T 1 + (1.37 - 3.88i)T
good2 1+(−1.34−1.74i)T+(−0.517+1.93i)T2 1 + (-1.34 - 1.74i)T + (-0.517 + 1.93i)T^{2}
5 1+(1.77+0.116i)T+(4.95+0.652i)T2 1 + (1.77 + 0.116i)T + (4.95 + 0.652i)T^{2}
7 1+(0.295+4.50i)T+(−6.94+0.913i)T2 1 + (0.295 + 4.50i)T + (-6.94 + 0.913i)T^{2}
11 1+(3.64+3.19i)T+(1.43+10.9i)T2 1 + (3.64 + 3.19i)T + (1.43 + 10.9i)T^{2}
13 1+(−3.24−0.868i)T+(11.2+6.5i)T2 1 + (-3.24 - 0.868i)T + (11.2 + 6.5i)T^{2}
19 1+(2.09+0.867i)T+(13.4+13.4i)T2 1 + (2.09 + 0.867i)T + (13.4 + 13.4i)T^{2}
23 1+(−1.20−3.54i)T+(−18.2+14.0i)T2 1 + (-1.20 - 3.54i)T + (-18.2 + 14.0i)T^{2}
29 1+(3.39−1.67i)T+(17.6−23.0i)T2 1 + (3.39 - 1.67i)T + (17.6 - 23.0i)T^{2}
31 1+(0.656+0.748i)T+(−4.04+30.7i)T2 1 + (0.656 + 0.748i)T + (-4.04 + 30.7i)T^{2}
37 1+(−0.0324−0.163i)T+(−34.1+14.1i)T2 1 + (-0.0324 - 0.163i)T + (-34.1 + 14.1i)T^{2}
41 1+(−1.80+3.66i)T+(−24.9−32.5i)T2 1 + (-1.80 + 3.66i)T + (-24.9 - 32.5i)T^{2}
43 1+(−1.31+9.95i)T+(−41.5−11.1i)T2 1 + (-1.31 + 9.95i)T + (-41.5 - 11.1i)T^{2}
47 1+(−7.98+2.13i)T+(40.7−23.5i)T2 1 + (-7.98 + 2.13i)T + (40.7 - 23.5i)T^{2}
53 1+(−1.75+4.24i)T+(−37.4−37.4i)T2 1 + (-1.75 + 4.24i)T + (-37.4 - 37.4i)T^{2}
59 1+(−2.19−1.68i)T+(15.2+56.9i)T2 1 + (-2.19 - 1.68i)T + (15.2 + 56.9i)T^{2}
61 1+(−7.16+0.469i)T+(60.4−7.96i)T2 1 + (-7.16 + 0.469i)T + (60.4 - 7.96i)T^{2}
67 1+(0.191−0.110i)T+(33.5−58.0i)T2 1 + (0.191 - 0.110i)T + (33.5 - 58.0i)T^{2}
71 1+(4.68−0.931i)T+(65.5−27.1i)T2 1 + (4.68 - 0.931i)T + (65.5 - 27.1i)T^{2}
73 1+(−3.14−2.10i)T+(27.9+67.4i)T2 1 + (-3.14 - 2.10i)T + (27.9 + 67.4i)T^{2}
79 1+(6.37−7.26i)T+(−10.3−78.3i)T2 1 + (6.37 - 7.26i)T + (-10.3 - 78.3i)T^{2}
83 1+(1.15−0.886i)T+(21.4−80.1i)T2 1 + (1.15 - 0.886i)T + (21.4 - 80.1i)T^{2}
89 1+(9.41−9.41i)T−89iT2 1 + (9.41 - 9.41i)T - 89iT^{2}
97 1+(−6.36−12.9i)T+(−59.0+76.9i)T2 1 + (-6.36 - 12.9i)T + (-59.0 + 76.9i)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

−13.49819384752753600786005489267, −13.04630675268674702309887578602, −11.02190294252189638541301632727, −10.41320055907150208388370585670, −8.590631675176297603400771034803, −7.83663769183486422010195616555, −7.03078174294476721286792599627, −5.50528668157711457566894745328, −4.01819634696692819216193209024, −3.73317585192419688665786205735, 2.20892965925012977300477370840, 3.02502752899459150251628265434, 4.47820235988990608119167343966, 5.87007600634914116647081719768, 7.58153242640271900547254047530, 8.626902147060216702763740201097, 9.759043581014818113832103424343, 11.15572210614106358119433290575, 11.98659294123619127387173580164, 12.70831129901527673612169031250

Graph of the ZZ-function along the critical line