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Properties

Label 2-1700-1700.1419-c0-0-1
Degree 22
Conductor 17001700
Sign −0.174+0.984i-0.174 + 0.984i
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  + (0.453 − 0.891i)2-s + (−0.587 − 0.809i)4-s + (0.987 − 0.156i)5-s + (−0.987 + 0.156i)8-s + (0.891 − 0.453i)9-s + (0.309 − 0.951i)10-s + (−0.0966 + 0.297i)13-s + (−0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s − 1.00i·18-s + (−0.707 − 0.707i)20-s + (0.951 − 0.309i)25-s + (0.221 + 0.221i)26-s + (−0.152 + 0.0366i)29-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.587 − 0.809i)4-s + (0.987 − 0.156i)5-s + (−0.987 + 0.156i)8-s + (0.891 − 0.453i)9-s + (0.309 − 0.951i)10-s + (−0.0966 + 0.297i)13-s + (−0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s − 1.00i·18-s + (−0.707 − 0.707i)20-s + (0.951 − 0.309i)25-s + (0.221 + 0.221i)26-s + (−0.152 + 0.0366i)29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

Λ(s)=(1700s/2 ΓC(s) L(s)=((−0.174+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1700s/2 ΓC(s) L(s)=((−0.174+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17001700    =    22⋅52⋅172^{2} \cdot 5^{2} \cdot 17
Sign: −0.174+0.984i-0.174 + 0.984i
Analytic conductor: 0.8484100.848410
Root analytic conductor: 0.9210920.921092
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1700(1419,⋅)\chi_{1700} (1419, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1700, ( :0), −0.174+0.984i)(2,\ 1700,\ (\ :0),\ -0.174 + 0.984i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5932427421.593242742
L(12)L(\frac12) ≈\approx 1.5932427421.593242742
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+(−0.453+0.891i)T 1 + (-0.453 + 0.891i)T
5 1+(−0.987+0.156i)T 1 + (-0.987 + 0.156i)T
17 1+(0.587+0.809i)T 1 + (0.587 + 0.809i)T
good3 1+(−0.891+0.453i)T2 1 + (-0.891 + 0.453i)T^{2}
7 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
11 1+(−0.156+0.987i)T2 1 + (-0.156 + 0.987i)T^{2}
13 1+(0.0966−0.297i)T+(−0.809−0.587i)T2 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2}
19 1+(−0.951+0.309i)T2 1 + (-0.951 + 0.309i)T^{2}
23 1+(0.156−0.987i)T2 1 + (0.156 - 0.987i)T^{2}
29 1+(0.152−0.0366i)T+(0.891−0.453i)T2 1 + (0.152 - 0.0366i)T + (0.891 - 0.453i)T^{2}
31 1+(−0.453+0.891i)T2 1 + (-0.453 + 0.891i)T^{2}
37 1+(−0.987+1.15i)T+(−0.156−0.987i)T2 1 + (-0.987 + 1.15i)T + (-0.156 - 0.987i)T^{2}
41 1+(−0.152−1.93i)T+(−0.987+0.156i)T2 1 + (-0.152 - 1.93i)T + (-0.987 + 0.156i)T^{2}
43 1+iT2 1 + iT^{2}
47 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
53 1+(1.95+0.309i)T+(0.951+0.309i)T2 1 + (1.95 + 0.309i)T + (0.951 + 0.309i)T^{2}
59 1+(−0.587+0.809i)T2 1 + (-0.587 + 0.809i)T^{2}
61 1+(0.303+0.355i)T+(−0.156+0.987i)T2 1 + (0.303 + 0.355i)T + (-0.156 + 0.987i)T^{2}
67 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
71 1+(0.891−0.453i)T2 1 + (0.891 - 0.453i)T^{2}
73 1+(1.70+0.133i)T+(0.987+0.156i)T2 1 + (1.70 + 0.133i)T + (0.987 + 0.156i)T^{2}
79 1+(−0.453−0.891i)T2 1 + (-0.453 - 0.891i)T^{2}
83 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
89 1+(−1.69+0.550i)T+(0.809−0.587i)T2 1 + (-1.69 + 0.550i)T + (0.809 - 0.587i)T^{2}
97 1+(−0.303−1.26i)T+(−0.891+0.453i)T2 1 + (-0.303 - 1.26i)T + (-0.891 + 0.453i)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

−9.423176910821703690616571344825, −9.059433605844372911409111699317, −7.74273397272427761140390819277, −6.58128023198851445401618098707, −6.06952841755594256055726173767, −4.91690457084515393840547557202, −4.42363164694028242177701296078, −3.20776033460349958947179185278, −2.21845851579484101191622725266, −1.22551918619632178116218428808, 1.79103417969873406581896297135, 2.99056750022948203419878567197, 4.17427498575721561342830314743, 4.93063924446196686762536207783, 5.80102841742334238504343603352, 6.48986184720219241804027051681, 7.21995757392721520004561431519, 8.039574310778653961293230283641, 8.863968407857058284230888777018, 9.665295013319071934665521676318

Graph of the ZZ-function along the critical line