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Properties

Label 2-1785-1.1-c1-0-38
Degree 22
Conductor 17851785
Sign −1-1
Analytic cond. 14.253214.2532
Root an. cond. 3.775353.77535
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 5-s + 7-s + 9-s + 2·11-s + 2·12-s − 5·13-s + 15-s + 4·16-s + 17-s + 2·19-s + 2·20-s − 21-s − 23-s + 25-s − 27-s − 2·28-s + 8·29-s + 31-s − 2·33-s − 35-s − 2·36-s − 3·37-s + 5·39-s − 7·41-s − 4·44-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s + 0.242·17-s + 0.458·19-s + 0.447·20-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 1.48·29-s + 0.179·31-s − 0.348·33-s − 0.169·35-s − 1/3·36-s − 0.493·37-s + 0.800·39-s − 1.09·41-s − 0.603·44-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: −1-1
Analytic conductor: 14.253214.2532
Root analytic conductor: 3.775353.77535
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1785, ( :1/2), −1)(2,\ 1785,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+pT2 1 + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+T+pT2 1 + T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1−T+pT2 1 - T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1+7T+pT2 1 + 7 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+T+pT2 1 + T + p T^{2}
53 1+8T+pT2 1 + 8 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1+6T+pT2 1 + 6 T + p T^{2}
83 1+13T+pT2 1 + 13 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1+6T+pT2 1 + 6 T + p T^{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

−8.875124152362556846665938617699, −8.130508438345186837574540048250, −7.36822758089724012978627819462, −6.50865430507889081738445221234, −5.38373779217897637905779872186, −4.79559864635169263531152538148, −4.11142735542990905129011155495, −2.98483251670407777110976435701, −1.33318755612277826546869106901, 0, 1.33318755612277826546869106901, 2.98483251670407777110976435701, 4.11142735542990905129011155495, 4.79559864635169263531152538148, 5.38373779217897637905779872186, 6.50865430507889081738445221234, 7.36822758089724012978627819462, 8.130508438345186837574540048250, 8.875124152362556846665938617699

Graph of the ZZ-function along the critical line