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Properties

Label 2-245-1.1-c1-0-2
Degree 22
Conductor 245245
Sign 11
Analytic cond. 1.956331.95633
Root an. cond. 1.398691.39869
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  − 1.41·2-s − 0.414·3-s + 5-s + 0.585·6-s + 2.82·8-s − 2.82·9-s − 1.41·10-s − 0.171·11-s + 4.41·13-s − 0.414·15-s − 4.00·16-s + 3.24·17-s + 4.00·18-s + 6·19-s + 0.242·22-s + 7.41·23-s − 1.17·24-s + 25-s − 6.24·26-s + 2.41·27-s − 8.65·29-s + 0.585·30-s + 10.2·31-s + 0.0710·33-s − 4.58·34-s + ⋯
L(s)  = 1  − 1.00·2-s − 0.239·3-s + 0.447·5-s + 0.239·6-s + 0.999·8-s − 0.942·9-s − 0.447·10-s − 0.0517·11-s + 1.22·13-s − 0.106·15-s − 1.00·16-s + 0.786·17-s + 0.942·18-s + 1.37·19-s + 0.0517·22-s + 1.54·23-s − 0.239·24-s + 0.200·25-s − 1.22·26-s + 0.464·27-s − 1.60·29-s + 0.106·30-s + 1.83·31-s + 0.0123·33-s − 0.786·34-s + ⋯

Functional equation

Λ(s)=(245s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(245s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 245245    =    5⋅725 \cdot 7^{2}
Sign: 11
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 245, ( :1/2), 1)(2,\ 245,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.70347377320.7034737732
L(12)L(\frac12) ≈\approx 0.70347377320.7034737732
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)
bad5 1−T 1 - T
7 1 1
good2 1+1.41T+2T2 1 + 1.41T + 2T^{2}
3 1+0.414T+3T2 1 + 0.414T + 3T^{2}
11 1+0.171T+11T2 1 + 0.171T + 11T^{2}
13 1−4.41T+13T2 1 - 4.41T + 13T^{2}
17 1−3.24T+17T2 1 - 3.24T + 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1−7.41T+23T2 1 - 7.41T + 23T^{2}
29 1+8.65T+29T2 1 + 8.65T + 29T^{2}
31 1−10.2T+31T2 1 - 10.2T + 31T^{2}
37 1−2.24T+37T2 1 - 2.24T + 37T^{2}
41 1+6.24T+41T2 1 + 6.24T + 41T^{2}
43 1−2T+43T2 1 - 2T + 43T^{2}
47 1+7.24T+47T2 1 + 7.24T + 47T^{2}
53 1−4.24T+53T2 1 - 4.24T + 53T^{2}
59 1+2.24T+59T2 1 + 2.24T + 59T^{2}
61 1+2.82T+61T2 1 + 2.82T + 61T^{2}
67 1+8.24T+67T2 1 + 8.24T + 67T^{2}
71 1+3.17T+71T2 1 + 3.17T + 71T^{2}
73 1+8.48T+73T2 1 + 8.48T + 73T^{2}
79 1−1.48T+79T2 1 - 1.48T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+8T+89T2 1 + 8T + 89T^{2}
97 1−13.2T+97T2 1 - 13.2T + 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

−11.73477395593774816396878547392, −11.00202302594167867959362115395, −10.03940226125010060606135121765, −9.154333733674727999064963391188, −8.411833973854258661558160498613, −7.37480225697226856105903711973, −6.01837265074468781325981483010, −5.00788806556701209132586036557, −3.22276589798217675804356448983, −1.16360101941472512386216658081, 1.16360101941472512386216658081, 3.22276589798217675804356448983, 5.00788806556701209132586036557, 6.01837265074468781325981483010, 7.37480225697226856105903711973, 8.411833973854258661558160498613, 9.154333733674727999064963391188, 10.03940226125010060606135121765, 11.00202302594167867959362115395, 11.73477395593774816396878547392

Graph of the ZZ-function along the critical line