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Properties

Label 1-300-300.71-r0-0-0
Degree $1$
Conductor $300$
Sign $-0.929 + 0.368i$
Analytic cond. $1.39319$
Root an. cond. $1.39319$
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  − 7-s + (−0.809 − 0.587i)11-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)23-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.809 + 0.587i)37-s + (0.809 − 0.587i)41-s − 43-s + (0.309 + 0.951i)47-s + 49-s + (−0.309 − 0.951i)53-s + (−0.809 + 0.587i)59-s + ⋯
L(s)  = 1  − 7-s + (−0.809 − 0.587i)11-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)23-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.809 + 0.587i)37-s + (0.809 − 0.587i)41-s − 43-s + (0.309 + 0.951i)47-s + 49-s + (−0.309 − 0.951i)53-s + (−0.809 + 0.587i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.929 + 0.368i$
Analytic conductor: \(1.39319\)
Root analytic conductor: \(1.39319\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 300,\ (0:\ ),\ -0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03926706968 + 0.2058451879i\)
\(L(\frac12)\) \(\approx\) \(0.03926706968 + 0.2058451879i\)
\(L(1)\) \(\approx\) \(0.6519498525 + 0.06586564388i\)
\(L(1)\) \(\approx\) \(0.6519498525 + 0.06586564388i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
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

−25.06963411138544231983486947400, −24.05912688577765347661741388617, −23.09314422543480153221558669449, −22.34342617762518754435363180163, −21.5245826644709017870980186705, −20.18836616169984764592659903997, −19.79844018022450078581514960240, −18.56159443244106809309010347006, −17.793611948987790785831873965710, −16.72589488468043137115504291004, −15.73417812590505537203573451369, −15.10715470091901487078611947990, −13.74518612751725299895316529715, −12.93370879325680026021805046510, −12.1369242093520169225796862471, −10.85260757535825954864655109560, −9.87595920907272074090070173086, −9.13926579995790251964582111595, −7.67449663664633135437784140404, −6.93953408156690245361187866078, −5.64780050451695938864429881578, −4.64297445497966664634012445880, −3.20794221004802115128672263340, −2.25534286602273722048272878357, −0.12421076659372906193632066592, 1.989674628569118491646590549049, 3.20779457345998553494177489687, 4.32877037691338478273057914305, 5.73096603622603057252817710577, 6.55337091280841884711051415504, 7.77915171531275978380878355755, 8.79554169796547434622782728792, 9.96080610090000950124595333862, 10.63316995807809364041672046462, 12.02826983279790294513547895975, 12.780860071119646392943897333188, 13.74410677524177807051657375227, 14.76385578750301259005646216581, 15.86887205482668471875300964542, 16.54625457539077954368199636474, 17.51857895716470358309409201793, 18.8198689233891534494839703804, 19.241516812328445910230181233769, 20.33495990684589206882107159252, 21.39987402752342491901694616540, 22.13043273045746032928303621376, 23.08669823084017557400585137539, 23.980704341807558578151358218678, 24.80533817755050501575063417893, 26.01946188665161814684136078985

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