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Properties

Label 1-405-405.218-r0-0-0
Degree $1$
Conductor $405$
Sign $-0.840 + 0.542i$
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  + (0.802 + 0.597i)2-s + (0.286 + 0.957i)4-s + (−0.727 + 0.686i)7-s + (−0.342 + 0.939i)8-s + (−0.893 − 0.448i)11-s + (0.918 + 0.396i)13-s + (−0.993 + 0.116i)14-s + (−0.835 + 0.549i)16-s + (−0.642 + 0.766i)17-s + (−0.766 + 0.642i)19-s + (−0.448 − 0.893i)22-s + (0.727 + 0.686i)23-s + (0.5 + 0.866i)26-s + (−0.866 − 0.5i)28-s + (−0.993 − 0.116i)29-s + ⋯
L(s)  = 1  + (0.802 + 0.597i)2-s + (0.286 + 0.957i)4-s + (−0.727 + 0.686i)7-s + (−0.342 + 0.939i)8-s + (−0.893 − 0.448i)11-s + (0.918 + 0.396i)13-s + (−0.993 + 0.116i)14-s + (−0.835 + 0.549i)16-s + (−0.642 + 0.766i)17-s + (−0.766 + 0.642i)19-s + (−0.448 − 0.893i)22-s + (0.727 + 0.686i)23-s + (0.5 + 0.866i)26-s + (−0.866 − 0.5i)28-s + (−0.993 − 0.116i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.840 + 0.542i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ -0.840 + 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4222529841 + 1.433372335i\)
\(L(\frac12)\) \(\approx\) \(0.4222529841 + 1.433372335i\)
\(L(1)\) \(\approx\) \(1.059840520 + 0.8102207968i\)
\(L(1)\) \(\approx\) \(1.059840520 + 0.8102207968i\)

Euler product

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

−23.70704089215641286956958352723, −22.914984115319856921577987719005, −22.53343927420159270822265210281, −21.24832362920538919560187610096, −20.58253299427559101282542352353, −19.91129222829437448069126573985, −18.91855997447262441646554217596, −18.134152966939609697660536535765, −16.86541561950538235829912908234, −15.7296770550824765099600933768, −15.254064552857024313913423032933, −13.93214541303410202589583674963, −13.1791150874506690499059748786, −12.7327035250402719201884814843, −11.36777598538092702916717491923, −10.63504590072340175169177715539, −9.87041835349671475710785565407, −8.69974753848441141044604329731, −7.172019960431321390280301557384, −6.42985740744432940264586042187, −5.21852413677626437363006394954, −4.28394236116333816563648575652, −3.220722667353028358688285319635, −2.2620274589963165598967819225, −0.6361246775257460512476201177, 2.10011080898884320091152810294, 3.22854676497995831616840740660, 4.13705901392356263969133416991, 5.51677305427973701756014577247, 6.09100557839160458111570579043, 7.09670945960391622730674718618, 8.348583497822758032383109295150, 8.95887635268792093826315945572, 10.50524053535705370162844329077, 11.44566184591802332937710900994, 12.57335561126347525235608408652, 13.16128543040549500611266274183, 13.99408285984554132648396746649, 15.27555771733231627599564904541, 15.65611561967596393293922381688, 16.57496701236313712513052534676, 17.50189869220100543738022808293, 18.6425708393664013276840794463, 19.36977685033447888539999750829, 20.8641928223049946011123146682, 21.24173912943627135560418373320, 22.24456838300579903793897287250, 23.00069372801118680207195036467, 23.80443138568657729179617101924, 24.55429280235028048067172299621

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