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Properties

Label 1-297-297.50-r0-0-0
Degree $1$
Conductor $297$
Sign $0.676 - 0.736i$
Analytic cond. $1.37926$
Root an. cond. $1.37926$
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  + (−0.997 − 0.0697i)2-s + (0.990 + 0.139i)4-s + (−0.559 − 0.829i)5-s + (0.882 + 0.469i)7-s + (−0.978 − 0.207i)8-s + (0.5 + 0.866i)10-s + (0.241 − 0.970i)13-s + (−0.848 − 0.529i)14-s + (0.961 + 0.275i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (−0.438 − 0.898i)20-s + (−0.173 − 0.984i)23-s + (−0.374 + 0.927i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (−0.997 − 0.0697i)2-s + (0.990 + 0.139i)4-s + (−0.559 − 0.829i)5-s + (0.882 + 0.469i)7-s + (−0.978 − 0.207i)8-s + (0.5 + 0.866i)10-s + (0.241 − 0.970i)13-s + (−0.848 − 0.529i)14-s + (0.961 + 0.275i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (−0.438 − 0.898i)20-s + (−0.173 − 0.984i)23-s + (−0.374 + 0.927i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.676 - 0.736i$
Analytic conductor: \(1.37926\)
Root analytic conductor: \(1.37926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (0:\ ),\ 0.676 - 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7296462583 - 0.3203453154i\)
\(L(\frac12)\) \(\approx\) \(0.7296462583 - 0.3203453154i\)
\(L(1)\) \(\approx\) \(0.7180056163 - 0.1411746288i\)
\(L(1)\) \(\approx\) \(0.7180056163 - 0.1411746288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.997 - 0.0697i)T \)
5 \( 1 + (-0.559 - 0.829i)T \)
7 \( 1 + (0.882 + 0.469i)T \)
13 \( 1 + (0.241 - 0.970i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.848 - 0.529i)T \)
31 \( 1 + (-0.719 - 0.694i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.848 + 0.529i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.990 + 0.139i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.615 - 0.788i)T \)
61 \( 1 + (0.719 - 0.694i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (0.997 + 0.0697i)T \)
83 \( 1 + (-0.241 - 0.970i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.559 - 0.829i)T \)
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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

−25.85016840492922829889821182458, −24.65815043678619714175892658362, −23.876927158845406693208817755208, −23.06018495200918319775881456416, −21.742239148403862581003549993381, −20.83717159380817682891036950724, −19.89122688823969100930185653813, −19.14257589899626476416587502113, −18.07975955303302353854066031054, −17.71585633621689947059172170054, −16.310660049068716638267321084030, −15.745290983066365093019882349940, −14.5261583766350736280636842176, −13.90697783233326364634953083417, −11.96146162132237660094049116652, −11.38352967488151072120654069273, −10.618994425063239589586874242603, −9.501375549454581810145542791018, −8.447434552898576169963482945269, −7.30540108978808581045674830608, −6.97116436722651465918353461581, −5.38231650817408488501731986489, −3.862988107767475218599260328709, −2.59417139333385462416087088141, −1.236971255166183194493127541946, 0.87168422264385070925181570113, 2.10068407667669990115992472807, 3.59589566731049507626241773392, 5.04409295483397154984827529515, 6.12130302258344409325461939662, 7.70431148752100217527290495122, 8.21076683432114472037264739283, 9.03066282272913315794320664782, 10.25445311101961007547042653397, 11.238671282078175228208670421599, 12.089887222729615537454277813647, 12.898931793004541498305407930317, 14.56607718272694138841761362987, 15.477197163937926433207444559, 16.19488022092969888933171833973, 17.26857929904544138108622171312, 17.93851292122821896042189688416, 18.916751106952927183942746034743, 19.88749125407307982565516101987, 20.603693070988223236588233371320, 21.28013672206225385992146137651, 22.59499512334148683026304160501, 23.95091145323261752958395114111, 24.47838378875564990792026900044, 25.21301500048741532842696262511

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