↑

Properties

Label 2-58e2-1.1-c1-0-8
Degree 22
Conductor 33643364
Sign 11
Analytic cond. 26.861626.8616
Root an. cond. 5.182825.18282
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.54·3-s + 2.86·5-s − 4.47·7-s + 3.45·9-s + 4.32·11-s − 6.10·13-s − 7.28·15-s + 3.82·17-s − 1.09·19-s + 11.3·21-s − 2.39·23-s + 3.21·25-s − 1.14·27-s − 3.71·31-s − 10.9·33-s − 12.8·35-s + 1.69·37-s + 15.5·39-s + 5.13·41-s − 2.00·43-s + 9.89·45-s − 9.28·47-s + 13.0·49-s − 9.70·51-s + 1.66·53-s + 12.3·55-s + 2.79·57-s + ⋯
L(s)  = 1  − 1.46·3-s + 1.28·5-s − 1.69·7-s + 1.15·9-s + 1.30·11-s − 1.69·13-s − 1.88·15-s + 0.927·17-s − 0.252·19-s + 2.48·21-s − 0.498·23-s + 0.643·25-s − 0.220·27-s − 0.666·31-s − 1.91·33-s − 2.16·35-s + 0.277·37-s + 2.48·39-s + 0.801·41-s − 0.305·43-s + 1.47·45-s − 1.35·47-s + 1.86·49-s − 1.35·51-s + 0.229·53-s + 1.67·55-s + 0.369·57-s + ⋯

Functional equation

Λ(s)=(3364s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3364s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33643364    =    22⋅2922^{2} \cdot 29^{2}
Sign: 11
Analytic conductor: 26.861626.8616
Root analytic conductor: 5.182825.18282
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3364, ( :1/2), 1)(2,\ 3364,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.88416245520.8841624552
L(12)L(\frac12) ≈\approx 0.88416245520.8841624552
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)
bad2 1 1
29 1 1
good3 1+2.54T+3T2 1 + 2.54T + 3T^{2}
5 1−2.86T+5T2 1 - 2.86T + 5T^{2}
7 1+4.47T+7T2 1 + 4.47T + 7T^{2}
11 1−4.32T+11T2 1 - 4.32T + 11T^{2}
13 1+6.10T+13T2 1 + 6.10T + 13T^{2}
17 1−3.82T+17T2 1 - 3.82T + 17T^{2}
19 1+1.09T+19T2 1 + 1.09T + 19T^{2}
23 1+2.39T+23T2 1 + 2.39T + 23T^{2}
31 1+3.71T+31T2 1 + 3.71T + 31T^{2}
37 1−1.69T+37T2 1 - 1.69T + 37T^{2}
41 1−5.13T+41T2 1 - 5.13T + 41T^{2}
43 1+2.00T+43T2 1 + 2.00T + 43T^{2}
47 1+9.28T+47T2 1 + 9.28T + 47T^{2}
53 1−1.66T+53T2 1 - 1.66T + 53T^{2}
59 1−0.0901T+59T2 1 - 0.0901T + 59T^{2}
61 1−8.52T+61T2 1 - 8.52T + 61T^{2}
67 1+8.77T+67T2 1 + 8.77T + 67T^{2}
71 1−9.23T+71T2 1 - 9.23T + 71T^{2}
73 1−2.10T+73T2 1 - 2.10T + 73T^{2}
79 1−5.42T+79T2 1 - 5.42T + 79T^{2}
83 1+6.23T+83T2 1 + 6.23T + 83T^{2}
89 1+3.62T+89T2 1 + 3.62T + 89T^{2}
97 1−6.64T+97T2 1 - 6.64T + 97T^{2}
show more
show less
   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

−9.012985559741126407320724744860, −7.53210742208176878762231083447, −6.68363130941481052388910357206, −6.38008158932146216004270864357, −5.71573406335475620562305462443, −5.12317868111834897608160373855, −4.06165568634943254846381034224, −2.99614549047729554833622527451, −1.88817865617250517774872943937, −0.58985551044840720427710827438, 0.58985551044840720427710827438, 1.88817865617250517774872943937, 2.99614549047729554833622527451, 4.06165568634943254846381034224, 5.12317868111834897608160373855, 5.71573406335475620562305462443, 6.38008158932146216004270864357, 6.68363130941481052388910357206, 7.53210742208176878762231083447, 9.012985559741126407320724744860

Graph of the ZZ-function along the critical line