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Properties

Label 2-2175-1.1-c1-0-18
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.772·2-s + 3-s − 1.40·4-s − 0.772·6-s − 2.17·7-s + 2.62·8-s + 9-s + 3·11-s − 1.40·12-s + 0.629·13-s + 1.68·14-s + 0.772·16-s − 4.17·17-s − 0.772·18-s + 4.80·19-s − 2.17·21-s − 2.31·22-s − 2.08·23-s + 2.62·24-s − 0.486·26-s + 27-s + 3.05·28-s − 29-s − 5.85·32-s + 3·33-s + 3.22·34-s − 1.40·36-s + ⋯
L(s)  = 1  − 0.546·2-s + 0.577·3-s − 0.701·4-s − 0.315·6-s − 0.822·7-s + 0.929·8-s + 0.333·9-s + 0.904·11-s − 0.404·12-s + 0.174·13-s + 0.449·14-s + 0.193·16-s − 1.01·17-s − 0.182·18-s + 1.10·19-s − 0.474·21-s − 0.494·22-s − 0.434·23-s + 0.536·24-s − 0.0954·26-s + 0.192·27-s + 0.576·28-s − 0.185·29-s − 1.03·32-s + 0.522·33-s + 0.553·34-s − 0.233·36-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 1.2288752501.228875250
L(12)L(\frac12) ≈\approx 1.2288752501.228875250
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+0.772T+2T2 1 + 0.772T + 2T^{2}
7 1+2.17T+7T2 1 + 2.17T + 7T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1−0.629T+13T2 1 - 0.629T + 13T^{2}
17 1+4.17T+17T2 1 + 4.17T + 17T^{2}
19 1−4.80T+19T2 1 - 4.80T + 19T^{2}
23 1+2.08T+23T2 1 + 2.08T + 23T^{2}
31 1+31T2 1 + 31T^{2}
37 1+6.08T+37T2 1 + 6.08T + 37T^{2}
41 1−0.824T+41T2 1 - 0.824T + 41T^{2}
43 1−8.72T+43T2 1 - 8.72T + 43T^{2}
47 1−8.98T+47T2 1 - 8.98T + 47T^{2}
53 1+6.88T+53T2 1 + 6.88T + 53T^{2}
59 1−6.45T+59T2 1 - 6.45T + 59T^{2}
61 1+2.80T+61T2 1 + 2.80T + 61T^{2}
67 1−11.0T+67T2 1 - 11.0T + 67T^{2}
71 1−2.63T+71T2 1 - 2.63T + 71T^{2}
73 1−14.7T+73T2 1 - 14.7T + 73T^{2}
79 1+12.0T+79T2 1 + 12.0T + 79T^{2}
83 1−7.62T+83T2 1 - 7.62T + 83T^{2}
89 1−17.8T+89T2 1 - 17.8T + 89T^{2}
97 1+0.538T+97T2 1 + 0.538T + 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

−9.202958411969937398615047864104, −8.509088257781598297406213354273, −7.63948407361398088025771493507, −6.92096912349969274890568152242, −6.04789831851620209930441203112, −4.95121189473724260167217003058, −3.99969796799729332953802914103, −3.39693749655534156302946259097, −2.06537309341979483712572173237, −0.78318140584315882525102430039, 0.78318140584315882525102430039, 2.06537309341979483712572173237, 3.39693749655534156302946259097, 3.99969796799729332953802914103, 4.95121189473724260167217003058, 6.04789831851620209930441203112, 6.92096912349969274890568152242, 7.63948407361398088025771493507, 8.509088257781598297406213354273, 9.202958411969937398615047864104

Graph of the ZZ-function along the critical line