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Properties

Label 1-1700-1700.1639-r0-0-0
Degree $1$
Conductor $1700$
Sign $0.360 - 0.932i$
Analytic cond. $7.89476$
Root an. cond. $7.89476$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.233 + 0.972i)3-s + (−0.923 − 0.382i)7-s + (−0.891 − 0.453i)9-s + (0.760 + 0.649i)11-s + (−0.951 + 0.309i)13-s + (−0.987 − 0.156i)19-s + (0.587 − 0.809i)21-s + (0.760 + 0.649i)23-s + (0.649 − 0.760i)27-s + (−0.972 − 0.233i)29-s + (0.852 + 0.522i)31-s + (−0.809 + 0.587i)33-s + (−0.760 + 0.649i)37-s + (−0.0784 − 0.996i)39-s + (0.0784 − 0.996i)41-s + ⋯
L(s)  = 1  + (−0.233 + 0.972i)3-s + (−0.923 − 0.382i)7-s + (−0.891 − 0.453i)9-s + (0.760 + 0.649i)11-s + (−0.951 + 0.309i)13-s + (−0.987 − 0.156i)19-s + (0.587 − 0.809i)21-s + (0.760 + 0.649i)23-s + (0.649 − 0.760i)27-s + (−0.972 − 0.233i)29-s + (0.852 + 0.522i)31-s + (−0.809 + 0.587i)33-s + (−0.760 + 0.649i)37-s + (−0.0784 − 0.996i)39-s + (0.0784 − 0.996i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.360 - 0.932i$
Analytic conductor: \(7.89476\)
Root analytic conductor: \(7.89476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1700,\ (0:\ ),\ 0.360 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3413216164 - 0.2340258218i\)
\(L(\frac12)\) \(\approx\) \(0.3413216164 - 0.2340258218i\)
\(L(1)\) \(\approx\) \(0.6914678248 + 0.1787922781i\)
\(L(1)\) \(\approx\) \(0.6914678248 + 0.1787922781i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.233 + 0.972i)T \)
7 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 + (0.760 + 0.649i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.987 - 0.156i)T \)
23 \( 1 + (0.760 + 0.649i)T \)
29 \( 1 + (-0.972 - 0.233i)T \)
31 \( 1 + (0.852 + 0.522i)T \)
37 \( 1 + (-0.760 + 0.649i)T \)
41 \( 1 + (0.0784 - 0.996i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.156 - 0.987i)T \)
59 \( 1 + (-0.891 - 0.453i)T \)
61 \( 1 + (0.649 - 0.760i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.233 - 0.972i)T \)
73 \( 1 + (-0.0784 - 0.996i)T \)
79 \( 1 + (0.852 - 0.522i)T \)
83 \( 1 + (-0.987 - 0.156i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
97 \( 1 + (0.972 + 0.233i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.1257460868349441452535080968, −19.62507398793495566718669790384, −18.85588897138095028162319193792, −18.59926818406715312137185408930, −17.300442995645648552594824060647, −16.98183866801327271985969310225, −16.24685026773965913342184687497, −15.10079228566983729597009621560, −14.53119176436091592169628802629, −13.55586989957347399010745381109, −12.906649202384899799217569557850, −12.32142450465148916652720316325, −11.62411557563932908312044110238, −10.7627435010386577694430037849, −9.829830209899615426717277443984, −8.8974226028720781158113471780, −8.3061188362397746379461302656, −7.20552259814825661331344326582, −6.62445326375024025436817202031, −5.93470835678968370695977818907, −5.12621831259580404617815328886, −3.87474026598842209394509444570, −2.872637869479108132578247460167, −2.18689622854453391135267640658, −0.96187442681448025595197468084, 0.17652049321092819175152676399, 1.77855478746982932395809979441, 2.97566142863318156808798710779, 3.72017547669794129139339222518, 4.540730698874300231681605585031, 5.209540653189232914800176073165, 6.44078402422867447080687302400, 6.81322584544963210666762781707, 7.98080768083807779701347073036, 9.18384265807631805078264917686, 9.49727292556913951510523288750, 10.252575970724770050461985668428, 11.01814743421726564472853835129, 11.91322815463330605669280981947, 12.56240754236552335518847904456, 13.499609912825210194782854910129, 14.43801911217661283738121700189, 15.04975798319414783179882105226, 15.70649345596263180267280909289, 16.585022357076112984078941658384, 17.17893616999068171814878236214, 17.537089881171102849448672067141, 19.02282428271426367670039081523, 19.47800038350975230800088492319, 20.18695669678544069869211137833

Graph of the $Z$-function along the critical line