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Properties

Label 2-1785-1.1-c1-0-39
Degree 22
Conductor 17851785
Sign 11
Analytic cond. 14.253214.2532
Root an. cond. 3.775353.77535
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  + 2.46·2-s − 3-s + 4.09·4-s + 5-s − 2.46·6-s − 7-s + 5.15·8-s + 9-s + 2.46·10-s + 5.09·11-s − 4.09·12-s − 0.461·13-s − 2.46·14-s − 15-s + 4.55·16-s − 17-s + 2.46·18-s + 2·19-s + 4.09·20-s + 21-s + 12.5·22-s − 1.15·23-s − 5.15·24-s + 25-s − 1.13·26-s − 27-s − 4.09·28-s + ⋯
L(s)  = 1  + 1.74·2-s − 0.577·3-s + 2.04·4-s + 0.447·5-s − 1.00·6-s − 0.377·7-s + 1.82·8-s + 0.333·9-s + 0.780·10-s + 1.53·11-s − 1.18·12-s − 0.128·13-s − 0.659·14-s − 0.258·15-s + 1.13·16-s − 0.242·17-s + 0.581·18-s + 0.458·19-s + 0.914·20-s + 0.218·21-s + 2.67·22-s − 0.240·23-s − 1.05·24-s + 0.200·25-s − 0.223·26-s − 0.192·27-s − 0.773·28-s + ⋯

Functional equation

Λ(s)=(1785s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1785s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17851785    =    3⋅5⋅7⋅173 \cdot 5 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 14.253214.2532
Root analytic conductor: 3.775353.77535
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1785, ( :1/2), 1)(2,\ 1785,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.7007905784.700790578
L(12)L(\frac12) ≈\approx 4.7007905784.700790578
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−T 1 - T
7 1+T 1 + T
17 1+T 1 + T
good2 1−2.46T+2T2 1 - 2.46T + 2T^{2}
11 1−5.09T+11T2 1 - 5.09T + 11T^{2}
13 1+0.461T+13T2 1 + 0.461T + 13T^{2}
19 1−2T+19T2 1 - 2T + 19T^{2}
23 1+1.15T+23T2 1 + 1.15T + 23T^{2}
29 1−8.24T+29T2 1 - 8.24T + 29T^{2}
31 1−4.29T+31T2 1 - 4.29T + 31T^{2}
37 1+2.23T+37T2 1 + 2.23T + 37T^{2}
41 1+4.30T+41T2 1 + 4.30T + 41T^{2}
43 1−5.33T+43T2 1 - 5.33T + 43T^{2}
47 1+7.38T+47T2 1 + 7.38T + 47T^{2}
53 1+0.691T+53T2 1 + 0.691T + 53T^{2}
59 1+3.73T+59T2 1 + 3.73T + 59T^{2}
61 1−8.96T+61T2 1 - 8.96T + 61T^{2}
67 1−3.15T+67T2 1 - 3.15T + 67T^{2}
71 1−8.29T+71T2 1 - 8.29T + 71T^{2}
73 1−2.70T+73T2 1 - 2.70T + 73T^{2}
79 1+4.93T+79T2 1 + 4.93T + 79T^{2}
83 1+15.4T+83T2 1 + 15.4T + 83T^{2}
89 1+8.58T+89T2 1 + 8.58T + 89T^{2}
97 1+6.58T+97T2 1 + 6.58T + 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

−9.511872168624075647253447211564, −8.434315458376606741783970360455, −7.05072153429717060336635983800, −6.57401551005486748842532159933, −6.02973105513635434249178378518, −5.14594943860019218434763631113, −4.39933390822749981525643999541, −3.61534940514463236847955847788, −2.61666086751245302536262113995, −1.35489160776684380839810096408, 1.35489160776684380839810096408, 2.61666086751245302536262113995, 3.61534940514463236847955847788, 4.39933390822749981525643999541, 5.14594943860019218434763631113, 6.02973105513635434249178378518, 6.57401551005486748842532159933, 7.05072153429717060336635983800, 8.434315458376606741783970360455, 9.511872168624075647253447211564

Graph of the ZZ-function along the critical line