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Properties

Label 2-1700-1700.147-c0-0-0
Degree 22
Conductor 17001700
Sign 0.853−0.521i0.853 - 0.521i
Analytic cond. 0.8484100.848410
Root an. cond. 0.9210920.921092
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.996 + 0.0784i)2-s + (0.987 − 0.156i)4-s + (−0.987 − 0.156i)5-s + (−0.972 + 0.233i)8-s + (−0.760 + 0.649i)9-s + (0.996 + 0.0784i)10-s + (0.444 − 0.144i)13-s + (0.951 − 0.309i)16-s + (0.453 − 0.891i)17-s + (0.707 − 0.707i)18-s − 20-s + (0.951 + 0.309i)25-s + (−0.431 + 0.178i)26-s + (0.666 + 1.80i)29-s + (−0.923 + 0.382i)32-s + ⋯
L(s)  = 1  + (−0.996 + 0.0784i)2-s + (0.987 − 0.156i)4-s + (−0.987 − 0.156i)5-s + (−0.972 + 0.233i)8-s + (−0.760 + 0.649i)9-s + (0.996 + 0.0784i)10-s + (0.444 − 0.144i)13-s + (0.951 − 0.309i)16-s + (0.453 − 0.891i)17-s + (0.707 − 0.707i)18-s − 20-s + (0.951 + 0.309i)25-s + (−0.431 + 0.178i)26-s + (0.666 + 1.80i)29-s + (−0.923 + 0.382i)32-s + ⋯

Functional equation

Λ(s)=(1700s/2 ΓC(s) L(s)=((0.853−0.521i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1700s/2 ΓC(s) L(s)=((0.853−0.521i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17001700    =    22⋅52⋅172^{2} \cdot 5^{2} \cdot 17
Sign: 0.853−0.521i0.853 - 0.521i
Analytic conductor: 0.8484100.848410
Root analytic conductor: 0.9210920.921092
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1700(147,⋅)\chi_{1700} (147, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1700, ( :0), 0.853−0.521i)(2,\ 1700,\ (\ :0),\ 0.853 - 0.521i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.55884181430.5588418143
L(12)L(\frac12) ≈\approx 0.55884181430.5588418143
L(1)L(1) 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)
bad2 1+(0.996−0.0784i)T 1 + (0.996 - 0.0784i)T
5 1+(0.987+0.156i)T 1 + (0.987 + 0.156i)T
17 1+(−0.453+0.891i)T 1 + (-0.453 + 0.891i)T
good3 1+(0.760−0.649i)T2 1 + (0.760 - 0.649i)T^{2}
7 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
11 1+(0.522+0.852i)T2 1 + (0.522 + 0.852i)T^{2}
13 1+(−0.444+0.144i)T+(0.809−0.587i)T2 1 + (-0.444 + 0.144i)T + (0.809 - 0.587i)T^{2}
19 1+(0.891−0.453i)T2 1 + (0.891 - 0.453i)T^{2}
23 1+(−0.852+0.522i)T2 1 + (-0.852 + 0.522i)T^{2}
29 1+(−0.666−1.80i)T+(−0.760+0.649i)T2 1 + (-0.666 - 1.80i)T + (-0.760 + 0.649i)T^{2}
31 1+(0.0784+0.996i)T2 1 + (0.0784 + 0.996i)T^{2}
37 1+(−0.226−0.0638i)T+(0.852+0.522i)T2 1 + (-0.226 - 0.0638i)T + (0.852 + 0.522i)T^{2}
41 1+(−1.56−1.23i)T+(0.233+0.972i)T2 1 + (-1.56 - 1.23i)T + (0.233 + 0.972i)T^{2}
43 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
47 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
53 1+(1.01+0.243i)T+(0.891+0.453i)T2 1 + (1.01 + 0.243i)T + (0.891 + 0.453i)T^{2}
59 1+(0.156−0.987i)T2 1 + (0.156 - 0.987i)T^{2}
61 1+(−1.41+0.398i)T+(0.852−0.522i)T2 1 + (-1.41 + 0.398i)T + (0.852 - 0.522i)T^{2}
67 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
71 1+(0.649+0.760i)T2 1 + (0.649 + 0.760i)T^{2}
73 1+(0.0984−0.831i)T+(−0.972−0.233i)T2 1 + (0.0984 - 0.831i)T + (-0.972 - 0.233i)T^{2}
79 1+(−0.0784+0.996i)T2 1 + (-0.0784 + 0.996i)T^{2}
83 1+(−0.453−0.891i)T2 1 + (-0.453 - 0.891i)T^{2}
89 1+(−0.0712−0.139i)T+(−0.587+0.809i)T2 1 + (-0.0712 - 0.139i)T + (-0.587 + 0.809i)T^{2}
97 1+(−1.42−0.657i)T+(0.649+0.760i)T2 1 + (-1.42 - 0.657i)T + (0.649 + 0.760i)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

−9.406728931324951987835948278363, −8.714003167785294719346158219335, −8.073979487627177811834951232829, −7.48646850885876466735922301423, −6.66902861432196667334661191163, −5.62772017522920335138554946750, −4.77389199071676007318147459860, −3.39358319281260119597218329807, −2.62724042075182796501868190859, −1.04185245266334702473969637005, 0.76218979166036468935683884055, 2.38818698831577730971186892906, 3.41898475360946198716131101417, 4.16711951501813260372940704529, 5.77254565491351866443424312169, 6.35532187709789006162917204146, 7.31533181073213775788686740261, 8.052612592380734033544348572721, 8.589254142191724018505262026097, 9.327168267297243099459266504453

Graph of the ZZ-function along the critical line