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Properties

Label 2-1700-1700.1379-c0-0-0
Degree 22
Conductor 17001700
Sign 0.548−0.836i0.548 - 0.836i
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 + (−0.465 + 1.93i)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 + (−0.465 + 1.93i)29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

Λ(s)=(1700s/2 ΓC(s) L(s)=((0.548−0.836i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1700s/2 ΓC(s) L(s)=((0.548−0.836i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17001700    =    22⋅52⋅172^{2} \cdot 5^{2} \cdot 17
Sign: 0.548−0.836i0.548 - 0.836i
Analytic conductor: 0.8484100.848410
Root analytic conductor: 0.9210920.921092
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1700(1379,⋅)\chi_{1700} (1379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1700, ( :0), 0.548−0.836i)(2,\ 1700,\ (\ :0),\ 0.548 - 0.836i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.35194673070.3519467307
L(12)L(\frac12) ≈\approx 0.35194673070.3519467307
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+(0.465−1.93i)T+(−0.891−0.453i)T2 1 + (0.465 - 1.93i)T + (-0.891 - 0.453i)T^{2}
31 1+(0.453+0.891i)T2 1 + (0.453 + 0.891i)T^{2}
37 1+(0.987−0.843i)T+(0.156−0.987i)T2 1 + (0.987 - 0.843i)T + (0.156 - 0.987i)T^{2}
41 1+(−0.465−0.0366i)T+(0.987+0.156i)T2 1 + (-0.465 - 0.0366i)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+(−1.47−1.26i)T+(0.156+0.987i)T2 1 + (-1.47 - 1.26i)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+(−0.0819−1.04i)T+(−0.987+0.156i)T2 1 + (-0.0819 - 1.04i)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+(1.47+0.355i)T+(0.891+0.453i)T2 1 + (1.47 + 0.355i)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

−9.581226485347804593251011181747, −8.623060959260584885713956278028, −8.577678570120816553634121292314, −7.46838254432190257764656601741, −6.67475812498951288873599306447, −5.39745687481286477607516260676, −4.39249716261579688567406160240, −3.60316739546623307757047060166, −2.83348283141836778375602404847, −1.39952828155611162408715615482, 0.32905139289381597737333822044, 2.31756797552344404347325239924, 3.63564105140008397873237180264, 4.62527979400434484125391586093, 5.41949051344950309480746225655, 6.31255523720394097898431700462, 7.17942795779257080686346366657, 7.87428009663507458000628982795, 8.409888367124078645557109801661, 9.185098844024206049701161742862

Graph of the ZZ-function along the critical line