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Properties

Label 1-405-405.68-r0-0-0
Degree $1$
Conductor $405$
Sign $-0.860 + 0.509i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ -0.860 + 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1484976303 + 0.5426975094i\)
\(L(\frac12)\) \(\approx\) \(0.1484976303 + 0.5426975094i\)
\(L(1)\) \(\approx\) \(0.5677567659 + 0.3011414098i\)
\(L(1)\) \(\approx\) \(0.5677567659 + 0.3011414098i\)

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

−24.10361362386469726867411984196, −23.06379923503337875430297139806, −22.03630026376413968011332469806, −21.04067396725087930279056404726, −20.56855009917833284759098330807, −19.66307248411057468687184479795, −18.65135847036034343696313153703, −18.05052674434539483252247424532, −16.98725592184498180738353401389, −16.495255799378405626622985825070, −15.26266047539316017129774841551, −14.1547707264071881205243350670, −13.14737764487833832875308060860, −12.18896432083288000340667955407, −11.29557473149284132886022348583, −10.34941634034489148189035977103, −9.82707564327056906846244887799, −8.31614845971689378627806860679, −7.887865846004559003568367445672, −6.859447022479149982210245778646, −5.30115653866991875812160106804, −4.15777323096005154603077709312, −2.940421396772922472485577914599, −1.879416087574081320121585548926, −0.41179945595950939092545481793, 1.654928351442686683479568920847, 2.54827918026250769883967896993, 4.55815583134634373336947563088, 5.391993215309500032555830505976, 6.398417439490760220890385700718, 7.658139532819536058373119008638, 8.14759091009233406683114242085, 9.3128188556228152269406910073, 10.09750204047206017881294144929, 11.12720872823633736109272082387, 12.04256185447591521949378026046, 13.241684984872373700411693468148, 14.58872972071446141362386368944, 15.01875401976241364286493339477, 15.90475972696474286724394394498, 17.023822203890635715488533182, 17.63399457179764130898758075124, 18.5291377433820512716017741017, 19.23142427583638895925347252082, 20.20579325378811297786367478773, 21.203808510254329983474805763017, 22.013536690352963290778712735998, 23.44894918092018187241120702452, 23.88436008326287100807279539310, 24.77259961284391456819738518933

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