↑

Properties

Label 2-2175-1.1-c1-0-35
Degree 22
Conductor 21752175
Sign 11
Analytic cond. 17.367417.3674
Root an. cond. 4.167424.16742
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  − 0.712·2-s + 3-s − 1.49·4-s − 0.712·6-s + 2.77·7-s + 2.48·8-s + 9-s + 4.26·11-s − 1.49·12-s − 0.779·13-s − 1.98·14-s + 1.21·16-s + 1.90·17-s − 0.712·18-s + 6.72·19-s + 2.77·21-s − 3.04·22-s − 2.17·23-s + 2.48·24-s + 0.555·26-s + 27-s − 4.14·28-s + 29-s − 8.82·31-s − 5.83·32-s + 4.26·33-s − 1.35·34-s + ⋯
L(s)  = 1  − 0.503·2-s + 0.577·3-s − 0.746·4-s − 0.290·6-s + 1.05·7-s + 0.879·8-s + 0.333·9-s + 1.28·11-s − 0.430·12-s − 0.216·13-s − 0.529·14-s + 0.302·16-s + 0.461·17-s − 0.167·18-s + 1.54·19-s + 0.606·21-s − 0.648·22-s − 0.452·23-s + 0.507·24-s + 0.108·26-s + 0.192·27-s − 0.783·28-s + 0.185·29-s − 1.58·31-s − 1.03·32-s + 0.742·33-s − 0.232·34-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 11
Analytic conductor: 17.367417.3674
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2175, ( :1/2), 1)(2,\ 2175,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8301611201.830161120
L(12)L(\frac12) ≈\approx 1.8301611201.830161120
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−T 1 - T
5 1 1
29 1−T 1 - T
good2 1+0.712T+2T2 1 + 0.712T + 2T^{2}
7 1−2.77T+7T2 1 - 2.77T + 7T^{2}
11 1−4.26T+11T2 1 - 4.26T + 11T^{2}
13 1+0.779T+13T2 1 + 0.779T + 13T^{2}
17 1−1.90T+17T2 1 - 1.90T + 17T^{2}
19 1−6.72T+19T2 1 - 6.72T + 19T^{2}
23 1+2.17T+23T2 1 + 2.17T + 23T^{2}
31 1+8.82T+31T2 1 + 8.82T + 31T^{2}
37 1+1.48T+37T2 1 + 1.48T + 37T^{2}
41 1+7.71T+41T2 1 + 7.71T + 41T^{2}
43 1−8.19T+43T2 1 - 8.19T + 43T^{2}
47 1−5.19T+47T2 1 - 5.19T + 47T^{2}
53 1−11.7T+53T2 1 - 11.7T + 53T^{2}
59 1+4.46T+59T2 1 + 4.46T + 59T^{2}
61 1+5.24T+61T2 1 + 5.24T + 61T^{2}
67 1−8.49T+67T2 1 - 8.49T + 67T^{2}
71 1−0.663T+71T2 1 - 0.663T + 71T^{2}
73 1+16.5T+73T2 1 + 16.5T + 73T^{2}
79 1−9.54T+79T2 1 - 9.54T + 79T^{2}
83 1−0.0123T+83T2 1 - 0.0123T + 83T^{2}
89 1+5.46T+89T2 1 + 5.46T + 89T^{2}
97 1−0.952T+97T2 1 - 0.952T + 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

−9.009662271731103879223555134214, −8.470988116287035332471923740060, −7.57137773425025841365339471799, −7.21220420017377197627456492370, −5.77226877713427626041308812698, −4.99143255628435602660286254547, −4.12370012665830086078976485887, −3.40326561960749779188572113345, −1.86287764034738179322724801509, −1.04047428153701103143638325675, 1.04047428153701103143638325675, 1.86287764034738179322724801509, 3.40326561960749779188572113345, 4.12370012665830086078976485887, 4.99143255628435602660286254547, 5.77226877713427626041308812698, 7.21220420017377197627456492370, 7.57137773425025841365339471799, 8.470988116287035332471923740060, 9.009662271731103879223555134214

Graph of the ZZ-function along the critical line