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Properties

Label 1-405-405.293-r0-0-0
Degree $1$
Conductor $405$
Sign $0.351 + 0.936i$
Analytic cond. $1.88081$
Root an. cond. $1.88081$
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  + (−0.727 − 0.686i)2-s + (0.0581 + 0.998i)4-s + (−0.448 − 0.893i)7-s + (0.642 − 0.766i)8-s + (−0.396 + 0.918i)11-s + (0.230 + 0.973i)13-s + (−0.286 + 0.957i)14-s + (−0.993 + 0.116i)16-s + (−0.984 + 0.173i)17-s + (−0.173 + 0.984i)19-s + (0.918 − 0.396i)22-s + (0.448 − 0.893i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.286 − 0.957i)29-s + ⋯
L(s)  = 1  + (−0.727 − 0.686i)2-s + (0.0581 + 0.998i)4-s + (−0.448 − 0.893i)7-s + (0.642 − 0.766i)8-s + (−0.396 + 0.918i)11-s + (0.230 + 0.973i)13-s + (−0.286 + 0.957i)14-s + (−0.993 + 0.116i)16-s + (−0.984 + 0.173i)17-s + (−0.173 + 0.984i)19-s + (0.918 − 0.396i)22-s + (0.448 − 0.893i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.286 − 0.957i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.351 + 0.936i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ 0.351 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3938285723 + 0.2726674783i\)
\(L(\frac12)\) \(\approx\) \(0.3938285723 + 0.2726674783i\)
\(L(1)\) \(\approx\) \(0.5984962651 - 0.05593121281i\)
\(L(1)\) \(\approx\) \(0.5984962651 - 0.05593121281i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.727 - 0.686i)T \)
7 \( 1 + (-0.448 - 0.893i)T \)
11 \( 1 + (-0.396 + 0.918i)T \)
13 \( 1 + (0.230 + 0.973i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.448 - 0.893i)T \)
29 \( 1 + (-0.286 - 0.957i)T \)
31 \( 1 + (-0.835 + 0.549i)T \)
37 \( 1 + (-0.342 + 0.939i)T \)
41 \( 1 + (0.686 + 0.727i)T \)
43 \( 1 + (-0.802 + 0.597i)T \)
47 \( 1 + (0.549 - 0.835i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.396 + 0.918i)T \)
61 \( 1 + (-0.0581 + 0.998i)T \)
67 \( 1 + (0.957 + 0.286i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.642 + 0.766i)T \)
79 \( 1 + (0.686 - 0.727i)T \)
83 \( 1 + (0.727 + 0.686i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.116 + 0.993i)T \)
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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

−24.30072877553065622652153190932, −23.601713446928234778822074446339, −22.45953781275772009926005588759, −21.76884520969588708144127909315, −20.476177654482101306579478274649, −19.57082023129642751461375127137, −18.83435181795113328665805604051, −18.00856228994305211562459058446, −17.30270066495626606790411462022, −16.01161029234973962527001988978, −15.67855419626809732566581834739, −14.78561830278613951525047153499, −13.56527736404266415041037734915, −12.789742296504969548401516846781, −11.256471388812193556140434252225, −10.71625317027318179554361115006, −9.323049197796993332748891854607, −8.85879038464219500936012584671, −7.82369998316506352106274145725, −6.75924256140328476110463274781, −5.760292639702807754074062704258, −5.10110169484552316087163041434, −3.28764794606946158404838492941, −2.09506748634241908284093880526, −0.358728186821822152167961501726, 1.41947405731050253424912019422, 2.49832400437487782530217033525, 3.851797198323492704570255510045, 4.55520741736445329171212762238, 6.48705037623592774090163418249, 7.2187912941704713658352571076, 8.26881351566020243883777627484, 9.31190456944696389927932883357, 10.149176286782512236031764335001, 10.86845492449157076909752748562, 11.907134990201663348645547157105, 12.87728637825752019118980359383, 13.55541767308410978018093909577, 14.82342082611604098168958593835, 16.09436893275291740582332488206, 16.75070266374910272567874913811, 17.58415332312208198966637086741, 18.520570331058093939567178781664, 19.28703523053832606910473666193, 20.24573567097499292155353009554, 20.721592872319782349403858876141, 21.75042867748840726774654881580, 22.739320701146703966460474651, 23.435225294419906769743492487775, 24.69898029243799476049466017787

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