↑

Properties

Label 1-133-133.102-r0-0-0
Degree $1$
Conductor $133$
Sign $-0.272 - 0.962i$
Analytic cond. $0.617649$
Root an. cond. $0.617649$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.272 - 0.962i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ -0.272 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6015429492 - 0.7957680141i\)
\(L(\frac12)\) \(\approx\) \(0.6015429492 - 0.7957680141i\)
\(L(1)\) \(\approx\) \(0.8228644545 - 0.5508237179i\)
\(L(1)\) \(\approx\) \(0.8228644545 - 0.5508237179i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.72490440529179224125166442412, −27.38735080145256678032733065060, −26.80682475867803323751691860000, −25.82355168783371884884008693544, −25.354367038839963046445128063203, −23.9983604269515762611030633137, −23.26848180544657174119181598030, −22.03743824489309764507474614856, −20.657244522826091053329421271548, −19.42928376361810735981092399795, −18.85399486628906140535768242940, −17.91799857156794200444963310188, −16.480799141247803957204059490229, −15.39570553900702289767645974844, −14.685150858642937869208353522948, −13.961741263523611261970928711336, −12.476082734700907638520473121790, −10.68027406171937773060722483925, −9.74023186944394033878578808362, −8.645435372809236258100109080900, −7.34214454756007315763841163028, −7.02399116019461504484624849634, −5.052353701131769137712136906682, −3.627400466298259039307524746949, −2.00333253912191074717003785934, 1.05812937445207455293763981757, 2.74584062607425030633615330019, 3.739435873592218209592593179698, 5.11283614765344999910570594340, 7.63008324323380934778024304690, 8.20983366447955981633456983991, 9.24861139102957565806346790120, 10.26361425892522246967863517230, 11.64178164721571162887660418886, 12.83098108419129345060325543049, 13.42092061533504662259384911460, 14.9393880563411469558021353214, 16.17293959081806927612857104001, 17.16494395169260946352099534843, 18.69917256190875899703874937150, 19.24828970629721777923281120421, 20.36303637465388910464754197841, 20.84145871465936983418673512745, 21.898412109445200128591127912477, 23.3517297533173545405041797327, 24.58792065296918213262725527560, 25.458459689097089784821102388327, 26.66737929892537705949472053194, 27.28235414560156290805267429458, 28.20357380641622893055390073585

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