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Properties

Label 1-469-469.277-r0-0-0
Degree $1$
Conductor $469$
Sign $0.500 + 0.865i$
Analytic cond. $2.17802$
Root an. cond. $2.17802$
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  + (−0.995 − 0.0950i)2-s + (0.0475 − 0.998i)3-s + (0.981 + 0.189i)4-s + (0.928 + 0.371i)5-s + (−0.142 + 0.989i)6-s + (−0.959 − 0.281i)8-s + (−0.995 − 0.0950i)9-s + (−0.888 − 0.458i)10-s + (−0.786 + 0.618i)11-s + (0.235 − 0.971i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)15-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.981 + 0.189i)18-s + (0.580 + 0.814i)19-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (0.0475 − 0.998i)3-s + (0.981 + 0.189i)4-s + (0.928 + 0.371i)5-s + (−0.142 + 0.989i)6-s + (−0.959 − 0.281i)8-s + (−0.995 − 0.0950i)9-s + (−0.888 − 0.458i)10-s + (−0.786 + 0.618i)11-s + (0.235 − 0.971i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)15-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.981 + 0.189i)18-s + (0.580 + 0.814i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.500 + 0.865i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.500 + 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5576984959 + 0.3216285521i\)
\(L(\frac12)\) \(\approx\) \(0.5576984959 + 0.3216285521i\)
\(L(1)\) \(\approx\) \(0.6740382424 + 0.02259111361i\)
\(L(1)\) \(\approx\) \(0.6740382424 + 0.02259111361i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.995 - 0.0950i)T \)
3 \( 1 + (0.0475 - 0.998i)T \)
5 \( 1 + (0.928 + 0.371i)T \)
11 \( 1 + (-0.786 + 0.618i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (-0.327 + 0.945i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
23 \( 1 + (-0.888 + 0.458i)T \)
29 \( 1 + T \)
31 \( 1 + (0.235 + 0.971i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (-0.888 + 0.458i)T \)
53 \( 1 + (0.981 + 0.189i)T \)
59 \( 1 + (0.235 + 0.971i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.928 - 0.371i)T \)
79 \( 1 + (0.235 - 0.971i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (0.0475 + 0.998i)T \)
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

−24.10891347977097467298376508421, −22.66827545780646819820016613507, −21.68429004070791608828560564793, −21.14827437085556568728812367953, −20.227210383672619622853134702, −19.69562843267081444942045015806, −18.272187765596261702028560662156, −17.733078864717695384529276677546, −16.76467665478173059099537730468, −16.17121326829458510819498943433, −15.38700157427941855453641722240, −14.34940291204178167501988754965, −13.42478891129536657549677419133, −12.02342036920485968289967387419, −11.0986326463728628681162111123, −10.0998161284519080390260524296, −9.698975519243822069528844281406, −8.7687135558199567035896226670, −7.9284939003768007234935063612, −6.590914728367815454830129767154, −5.50998144253123690642432584255, −4.81892883151235308550902056040, −2.977766464155258549643915946471, −2.31006416916902355365572074766, −0.456120489694097314496990540297, 1.539593356044893550098072665147, 2.15952421210965657079753115742, 3.16194407560105389621326465616, 5.27299135287978231040233832226, 6.28992478879503970692876533427, 7.04990859915688192168994901049, 7.89057629492126231086021484079, 8.80955588363214260436082971802, 10.014881096388396710690486631726, 10.43612358539022407245469392330, 11.8563581790573471683711413561, 12.42645228708061199385655889727, 13.51856134986609659397861946419, 14.44315631951937736973807819377, 15.39733038796151357560913817204, 16.67615887282429063716393824891, 17.50590552007111773111812541701, 17.94768875783994944451072362241, 18.732407550923917751024003225216, 19.55537440284624896782602596134, 20.34199056661810464798325346190, 21.30425506372993493332608408843, 22.184062802909624748022246033244, 23.40088113614345291774635857998, 24.291506033583002926178146075375

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