↑

Properties

Label 1-165-165.74-r0-0-0
Degree $1$
Conductor $165$
Sign $0.970 - 0.242i$
Analytic cond. $0.766256$
Root an. cond. $0.766256$
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  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + 23-s + (0.809 + 0.587i)26-s + (0.309 + 0.951i)28-s + (−0.809 − 0.587i)29-s + (0.309 − 0.951i)31-s − 32-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + 23-s + (0.809 + 0.587i)26-s + (0.309 + 0.951i)28-s + (−0.809 − 0.587i)29-s + (0.309 − 0.951i)31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(0.766256\)
Root analytic conductor: \(0.766256\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 165,\ (0:\ ),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7379867646 - 0.09071811885i\)
\(L(\frac12)\) \(\approx\) \(0.7379867646 - 0.09071811885i\)
\(L(1)\) \(\approx\) \(0.7688377863 + 0.1153042120i\)
\(L(1)\) \(\approx\) \(0.7688377863 + 0.1153042120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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

−27.981690026003707697858102710427, −26.79041320401558427401341414966, −26.05296426809650742828760930305, −25.0545417024269959268988226706, −23.62437109800429800627139616732, −22.60225008729959579802418604854, −21.76778786785575486188065092994, −20.93545828641962968545151219880, −19.75639640304738599618549487407, −18.99356411051259148351358161770, −18.20973020513709469920817563508, −16.959596320209341991967809138655, −16.03998340024626090607360019177, −14.59234387170740884455499616238, −13.37049601683250019962709924997, −12.51339384284185882523210817346, −11.55111557171740289784897059148, −10.44204804995668454725146758525, −9.33238489900858621756621705981, −8.60241715010725354146859351532, −7.0639753466477982926753458283, −5.59232494331384478901559375041, −4.06747166719045205367795732300, −2.97402529152534057978652593513, −1.58692449802418024822625111297, 0.744447146967025827803393591691, 3.13718207389326706326724007799, 4.61286866416055944335337951260, 5.839028950727689038838277126729, 6.95473416582259038967757609163, 7.832826477281980364673664548744, 9.21224499892419934393915558517, 9.98769011530586383232483680147, 11.25324540813799754487729888872, 13.075993639419330570611332415509, 13.55787074866454528872501256883, 14.94354642903221028440798309198, 15.81750186782543367118852501361, 16.685378731888821662433888760733, 17.66417433961549952995089218670, 18.62459505482432808400226696115, 19.66867608140720804452675991213, 20.62945636129972869248857176784, 22.429787030292563141754106354810, 22.73674560414115181564945193129, 23.90339125546777321093456307346, 24.87585528108367109965547369520, 25.70387611039328213679292109415, 26.59653528891709325314228120881, 27.36642839183428495493191368597

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