Properties

Label 1-643-643.642-r1-0-0
Degree $1$
Conductor $643$
Sign $1$
Analytic cond. $69.0999$
Root an. cond. $69.0999$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $1$
Analytic conductor: \(69.0999\)
Root analytic conductor: \(69.0999\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{643} (642, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3025150866\)
\(L(\frac12)\) \(\approx\) \(0.3025150866\)
\(L(1)\) \(\approx\) \(0.3716769605\)
\(L(1)\) \(\approx\) \(0.3716769605\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
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

−22.93252018227737288288950626970, −21.63008185259395885453904545478, −21.077041281570181464566961900623, −20.0698106821699845451439030240, −19.18876002554516113675405608004, −18.51276375916277736284833691126, −17.62483245169449670032629637073, −17.138759356440639648053180526342, −16.206666004260501229182964193220, −15.32516739261201681060424703351, −14.96319213777947410935107690987, −13.18511755201919185607738244326, −12.080432429516668260029980702475, −11.67530764723701981668610388031, −10.6505964850685370141247242711, −10.37851599722103295335148798721, −8.83825019532104945091702563189, −8.0675998100212648622853035252, −7.250042550799442389151647718, −6.530401532291666613576745422887, −5.08602076304949319475866768162, −4.52350583466684873628270372338, −2.84231951528713338023970079854, −1.642687838760410474121242296788, −0.34247156348446754602994361323, 0.34247156348446754602994361323, 1.642687838760410474121242296788, 2.84231951528713338023970079854, 4.52350583466684873628270372338, 5.08602076304949319475866768162, 6.530401532291666613576745422887, 7.250042550799442389151647718, 8.0675998100212648622853035252, 8.83825019532104945091702563189, 10.37851599722103295335148798721, 10.6505964850685370141247242711, 11.67530764723701981668610388031, 12.080432429516668260029980702475, 13.18511755201919185607738244326, 14.96319213777947410935107690987, 15.32516739261201681060424703351, 16.206666004260501229182964193220, 17.138759356440639648053180526342, 17.62483245169449670032629637073, 18.51276375916277736284833691126, 19.18876002554516113675405608004, 20.0698106821699845451439030240, 21.077041281570181464566961900623, 21.63008185259395885453904545478, 22.93252018227737288288950626970

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