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Properties

Label 1-1617-1617.425-r1-0-0
Degree $1$
Conductor $1617$
Sign $-0.262 + 0.964i$
Analytic cond. $173.770$
Root an. cond. $173.770$
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  + (−0.873 − 0.486i)2-s + (0.525 + 0.850i)4-s + (−0.772 − 0.635i)5-s + (−0.0448 − 0.998i)8-s + (0.365 + 0.930i)10-s + (0.858 − 0.512i)13-s + (−0.447 + 0.894i)16-s + (0.280 + 0.959i)17-s + (−0.104 + 0.994i)19-s + (0.134 − 0.990i)20-s + (−0.826 − 0.563i)23-s + (0.193 + 0.981i)25-s + (−0.999 + 0.0299i)26-s + (0.983 − 0.178i)29-s + (0.978 + 0.207i)31-s + (0.826 − 0.563i)32-s + ⋯
L(s)  = 1  + (−0.873 − 0.486i)2-s + (0.525 + 0.850i)4-s + (−0.772 − 0.635i)5-s + (−0.0448 − 0.998i)8-s + (0.365 + 0.930i)10-s + (0.858 − 0.512i)13-s + (−0.447 + 0.894i)16-s + (0.280 + 0.959i)17-s + (−0.104 + 0.994i)19-s + (0.134 − 0.990i)20-s + (−0.826 − 0.563i)23-s + (0.193 + 0.981i)25-s + (−0.999 + 0.0299i)26-s + (0.983 − 0.178i)29-s + (0.978 + 0.207i)31-s + (0.826 − 0.563i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.262 + 0.964i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ -0.262 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2279242877 + 0.2983311172i\)
\(L(\frac12)\) \(\approx\) \(0.2279242877 + 0.2983311172i\)
\(L(1)\) \(\approx\) \(0.5948450059 - 0.1202857913i\)
\(L(1)\) \(\approx\) \(0.5948450059 - 0.1202857913i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.873 - 0.486i)T \)
5 \( 1 + (-0.772 - 0.635i)T \)
13 \( 1 + (0.858 - 0.512i)T \)
17 \( 1 + (0.280 + 0.959i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.826 - 0.563i)T \)
29 \( 1 + (0.983 - 0.178i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.646 - 0.762i)T \)
41 \( 1 + (0.0448 + 0.998i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.992 - 0.119i)T \)
53 \( 1 + (-0.971 + 0.237i)T \)
59 \( 1 + (-0.842 - 0.538i)T \)
61 \( 1 + (0.971 + 0.237i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.134 - 0.990i)T \)
73 \( 1 + (0.992 + 0.119i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.858 - 0.512i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−19.913530397857922130891661517870, −19.00043887321973546625132101396, −18.69718883079312848906087483382, −17.830593024084431023695644665853, −17.199814672003115730847851755, −16.060177261796374982591537819411, −15.78093992782767343543162170422, −15.127315771949509030645261887959, −14.02006340144071791494727063263, −13.72700120551070047198423244509, −12.07300036859199977125192234560, −11.611699388722923099358654566861, −10.81980437316320144679706835209, −10.13621472121786533379190383858, −9.20106875913478366415890286076, −8.45387490130232822595849821927, −7.7289177599660763257541598356, −6.88908911944713637040341132094, −6.424033750286907284907916370011, −5.30154231265447098417487014897, −4.33130968367041764376328073204, −3.211199413141284897435891593036, −2.317515909836481791418645553955, −1.08744966783692843901221390684, −0.12159604164668954908238309341, 0.95160269278079429260990696204, 1.66607762009248527800866989422, 2.96247319541865084637590597027, 3.77635656870654940322344791760, 4.46396255054283493113178826182, 5.83467067978950533190931107014, 6.61791023767573811540523163724, 7.8890477521624185257285555978, 8.16586736490905390238082182337, 8.82001950710534551071050072379, 9.92158956432658522862138206899, 10.522845272856848143180232500520, 11.32749322212105273978157543953, 12.21977183919135882738311917667, 12.55556876191590022845515755691, 13.4845038094570819920158404105, 14.61547268119455326497229769120, 15.65678924715529667741304471401, 16.04196026985032807565473944411, 16.83848325187355853333380919930, 17.53261069821312062350798237659, 18.37111639088758322942957878321, 19.05637376339190048803328769256, 19.712990208795995897665458094897, 20.35771044218955779049648716735

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