Properties

Label 1-433-433.432-r0-0-0
Degree $1$
Conductor $433$
Sign $1$
Analytic cond. $2.01084$
Root an. cond. $2.01084$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 14-s − 15-s + 16-s + 17-s + 18-s − 19-s − 20-s − 21-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 14-s − 15-s + 16-s + 17-s + 18-s − 19-s − 20-s − 21-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 433 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 433 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(433\)
Sign: $1$
Analytic conductor: \(2.01084\)
Root analytic conductor: \(2.01084\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{433} (432, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 433,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.030983008\)
\(L(\frac12)\) \(\approx\) \(3.030983008\)
\(L(1)\) \(\approx\) \(2.248559241\)
\(L(1)\) \(\approx\) \(2.248559241\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad433 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−23.9628653075227899591358606395, −23.29050822445761714108211338282, −22.46803498295939339202032202738, −21.585961609948152102161509374620, −20.61121836717329214417598856460, −19.84475601912085912322235716913, −19.36052009654573191926899231641, −18.5093074717930894089062411474, −16.42899318797128457638069970821, −16.283492755674939065797098587718, −15.0383633364175083945575937397, −14.66453698581029335488121331417, −13.5318689036573867972681920324, −12.79231520010871830473934118486, −12.020734429176987338623275995525, −10.96010369218615935681463108293, −9.83311838660726548284811305624, −8.69343829858989370531920831481, −7.70917014884682547292481007016, −6.81876397006086288370325344878, −5.89599391772281396490634327968, −4.119232788649975510780846373364, −3.81061602141632160056488865861, −2.92319151802462182947924685839, −1.502348492330017536290963408379, 1.502348492330017536290963408379, 2.92319151802462182947924685839, 3.81061602141632160056488865861, 4.119232788649975510780846373364, 5.89599391772281396490634327968, 6.81876397006086288370325344878, 7.70917014884682547292481007016, 8.69343829858989370531920831481, 9.83311838660726548284811305624, 10.96010369218615935681463108293, 12.020734429176987338623275995525, 12.79231520010871830473934118486, 13.5318689036573867972681920324, 14.66453698581029335488121331417, 15.0383633364175083945575937397, 16.283492755674939065797098587718, 16.42899318797128457638069970821, 18.5093074717930894089062411474, 19.36052009654573191926899231641, 19.84475601912085912322235716913, 20.61121836717329214417598856460, 21.585961609948152102161509374620, 22.46803498295939339202032202738, 23.29050822445761714108211338282, 23.9628653075227899591358606395

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