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Properties

Label 2-29e2-1.1-c1-0-52
Degree 22
Conductor 841841
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.80·2-s − 0.445·3-s + 1.24·4-s + 0.356·5-s − 0.801·6-s − 4.04·7-s − 1.35·8-s − 2.80·9-s + 0.643·10-s − 2.91·11-s − 0.554·12-s + 5.18·13-s − 7.29·14-s − 0.158·15-s − 4.93·16-s − 1.10·17-s − 5.04·18-s − 2.04·19-s + 0.445·20-s + 1.80·21-s − 5.24·22-s − 4.13·23-s + 0.603·24-s − 4.87·25-s + 9.34·26-s + 2.58·27-s − 5.04·28-s + ⋯
L(s)  = 1  + 1.27·2-s − 0.256·3-s + 0.623·4-s + 0.159·5-s − 0.327·6-s − 1.53·7-s − 0.479·8-s − 0.933·9-s + 0.203·10-s − 0.877·11-s − 0.160·12-s + 1.43·13-s − 1.94·14-s − 0.0410·15-s − 1.23·16-s − 0.269·17-s − 1.19·18-s − 0.470·19-s + 0.0995·20-s + 0.393·21-s − 1.11·22-s − 0.862·23-s + 0.123·24-s − 0.974·25-s + 1.83·26-s + 0.496·27-s − 0.954·28-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: −1-1
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: 11
Selberg data: (2, 841, ( :1/2), −1)(2,\ 841,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−1.80T+2T2 1 - 1.80T + 2T^{2}
3 1+0.445T+3T2 1 + 0.445T + 3T^{2}
5 1−0.356T+5T2 1 - 0.356T + 5T^{2}
7 1+4.04T+7T2 1 + 4.04T + 7T^{2}
11 1+2.91T+11T2 1 + 2.91T + 11T^{2}
13 1−5.18T+13T2 1 - 5.18T + 13T^{2}
17 1+1.10T+17T2 1 + 1.10T + 17T^{2}
19 1+2.04T+19T2 1 + 2.04T + 19T^{2}
23 1+4.13T+23T2 1 + 4.13T + 23T^{2}
31 1−6.35T+31T2 1 - 6.35T + 31T^{2}
37 1−2.91T+37T2 1 - 2.91T + 37T^{2}
41 1+0.396T+41T2 1 + 0.396T + 41T^{2}
43 1+5.74T+43T2 1 + 5.74T + 43T^{2}
47 1−7.80T+47T2 1 - 7.80T + 47T^{2}
53 1+4.35T+53T2 1 + 4.35T + 53T^{2}
59 1+9.10T+59T2 1 + 9.10T + 59T^{2}
61 1−6.04T+61T2 1 - 6.04T + 61T^{2}
67 1−0.374T+67T2 1 - 0.374T + 67T^{2}
71 1+11.4T+71T2 1 + 11.4T + 71T^{2}
73 1−8.94T+73T2 1 - 8.94T + 73T^{2}
79 1−0.594T+79T2 1 - 0.594T + 79T^{2}
83 1+9.43T+83T2 1 + 9.43T + 83T^{2}
89 1+1.42T+89T2 1 + 1.42T + 89T^{2}
97 1+15.7T+97T2 1 + 15.7T + 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

−9.888248606523496886580597135121, −8.950124711432689622074153127905, −8.107137431893207583710431436740, −6.57056055058688933030254316122, −6.09688110311587608278971942747, −5.52908969738945855010800836708, −4.24688492911943017040765548776, −3.36770041771616000057194031009, −2.56346244963340507972511034433, 0, 2.56346244963340507972511034433, 3.36770041771616000057194031009, 4.24688492911943017040765548776, 5.52908969738945855010800836708, 6.09688110311587608278971942747, 6.57056055058688933030254316122, 8.107137431893207583710431436740, 8.950124711432689622074153127905, 9.888248606523496886580597135121

Graph of the ZZ-function along the critical line