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Properties

Label 2-29e2-1.1-c1-0-2
Degree 22
Conductor 841841
Sign 11
Analytic cond. 6.715416.71541
Root an. cond. 2.591412.59141
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.171·2-s − 1.12·3-s − 1.97·4-s − 2.82·5-s − 0.192·6-s − 2.92·7-s − 0.679·8-s − 1.73·9-s − 0.483·10-s − 3.99·11-s + 2.21·12-s + 2.81·13-s − 0.500·14-s + 3.17·15-s + 3.82·16-s + 0.482·17-s − 0.296·18-s − 3.04·19-s + 5.57·20-s + 3.28·21-s − 0.682·22-s + 5.51·23-s + 0.763·24-s + 2.99·25-s + 0.480·26-s + 5.32·27-s + 5.76·28-s + ⋯
L(s)  = 1  + 0.120·2-s − 0.649·3-s − 0.985·4-s − 1.26·5-s − 0.0784·6-s − 1.10·7-s − 0.240·8-s − 0.578·9-s − 0.152·10-s − 1.20·11-s + 0.639·12-s + 0.779·13-s − 0.133·14-s + 0.820·15-s + 0.956·16-s + 0.117·17-s − 0.0699·18-s − 0.699·19-s + 1.24·20-s + 0.717·21-s − 0.145·22-s + 1.14·23-s + 0.155·24-s + 0.598·25-s + 0.0942·26-s + 1.02·27-s + 1.08·28-s + ⋯

Functional equation

Λ(s)=(841s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(841s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 841841    =    29229^{2}
Sign: 11
Analytic conductor: 6.715416.71541
Root analytic conductor: 2.591412.59141
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 841, ( :1/2), 1)(2,\ 841,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.26309731390.2630973139
L(12)L(\frac12) ≈\approx 0.26309731390.2630973139
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)
bad29 1 1
good2 1−0.171T+2T2 1 - 0.171T + 2T^{2}
3 1+1.12T+3T2 1 + 1.12T + 3T^{2}
5 1+2.82T+5T2 1 + 2.82T + 5T^{2}
7 1+2.92T+7T2 1 + 2.92T + 7T^{2}
11 1+3.99T+11T2 1 + 3.99T + 11T^{2}
13 1−2.81T+13T2 1 - 2.81T + 13T^{2}
17 1−0.482T+17T2 1 - 0.482T + 17T^{2}
19 1+3.04T+19T2 1 + 3.04T + 19T^{2}
23 1−5.51T+23T2 1 - 5.51T + 23T^{2}
31 1+3.85T+31T2 1 + 3.85T + 31T^{2}
37 1+11.5T+37T2 1 + 11.5T + 37T^{2}
41 1+5.10T+41T2 1 + 5.10T + 41T^{2}
43 1−8.21T+43T2 1 - 8.21T + 43T^{2}
47 1−2.38T+47T2 1 - 2.38T + 47T^{2}
53 1+0.446T+53T2 1 + 0.446T + 53T^{2}
59 1−1.24T+59T2 1 - 1.24T + 59T^{2}
61 1+8.59T+61T2 1 + 8.59T + 61T^{2}
67 1+0.944T+67T2 1 + 0.944T + 67T^{2}
71 1−5.97T+71T2 1 - 5.97T + 71T^{2}
73 1−0.483T+73T2 1 - 0.483T + 73T^{2}
79 1−1.85T+79T2 1 - 1.85T + 79T^{2}
83 1−4.35T+83T2 1 - 4.35T + 83T^{2}
89 1−15.5T+89T2 1 - 15.5T + 89T^{2}
97 1−2.41T+97T2 1 - 2.41T + 97T^{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

−10.47905412121954092065970427304, −9.192249751753887083184060426213, −8.562178649225386819996564275847, −7.74294946136607039435436977133, −6.66888815720432359000066946431, −5.65733387505460032816084082833, −4.90722893189871953068718608086, −3.76834889048235215287294082997, −3.07870985183737047725896424491, −0.40020865405981684223286852076, 0.40020865405981684223286852076, 3.07870985183737047725896424491, 3.76834889048235215287294082997, 4.90722893189871953068718608086, 5.65733387505460032816084082833, 6.66888815720432359000066946431, 7.74294946136607039435436977133, 8.562178649225386819996564275847, 9.192249751753887083184060426213, 10.47905412121954092065970427304

Graph of the ZZ-function along the critical line