↑

Properties

Label 1-164-164.47-r0-0-0
Degree $1$
Conductor $164$
Sign $-0.601 - 0.799i$
Analytic cond. $0.761612$
Root an. cond. $0.761612$
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  + (0.707 − 0.707i)3-s + (−0.951 − 0.309i)5-s + (−0.987 + 0.156i)7-s − i·9-s + (−0.891 − 0.453i)11-s + (0.156 − 0.987i)13-s + (−0.891 + 0.453i)15-s + (0.453 − 0.891i)17-s + (−0.156 − 0.987i)19-s + (−0.587 + 0.809i)21-s + (−0.809 + 0.587i)23-s + (0.809 + 0.587i)25-s + (−0.707 − 0.707i)27-s + (0.453 + 0.891i)29-s + (0.309 + 0.951i)31-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.951 − 0.309i)5-s + (−0.987 + 0.156i)7-s − i·9-s + (−0.891 − 0.453i)11-s + (0.156 − 0.987i)13-s + (−0.891 + 0.453i)15-s + (0.453 − 0.891i)17-s + (−0.156 − 0.987i)19-s + (−0.587 + 0.809i)21-s + (−0.809 + 0.587i)23-s + (0.809 + 0.587i)25-s + (−0.707 − 0.707i)27-s + (0.453 + 0.891i)29-s + (0.309 + 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.601 - 0.799i$
Analytic conductor: \(0.761612\)
Root analytic conductor: \(0.761612\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 164,\ (0:\ ),\ -0.601 - 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3736267070 - 0.7485747549i\)
\(L(\frac12)\) \(\approx\) \(0.3736267070 - 0.7485747549i\)
\(L(1)\) \(\approx\) \(0.7973620499 - 0.4351232574i\)
\(L(1)\) \(\approx\) \(0.7973620499 - 0.4351232574i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (-0.987 + 0.156i)T \)
11 \( 1 + (-0.891 - 0.453i)T \)
13 \( 1 + (0.156 - 0.987i)T \)
17 \( 1 + (0.453 - 0.891i)T \)
19 \( 1 + (-0.156 - 0.987i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (-0.987 - 0.156i)T \)
53 \( 1 + (-0.453 - 0.891i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.587 - 0.809i)T \)
67 \( 1 + (0.891 - 0.453i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (0.987 - 0.156i)T \)
97 \( 1 + (0.891 - 0.453i)T \)
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

−27.99191903068710356511855491787, −26.85472682317764229883487379317, −26.161331947011061017425580668782, −25.59126361508528943109263117247, −24.078432125715274814292838631950, −23.10553465992689140806713375017, −22.27435690064585768716775793870, −21.103224858565962085008931498825, −20.242301572325357591463218658889, −19.19421342222321922281620750368, −18.7163719472694186148065306908, −16.78205698139034262104794241325, −16.00889531984771133150604498846, −15.2259455307246478235633313799, −14.242476533280581983955130796845, −13.04977126214555220519941364262, −11.89853302700672961272291905287, −10.479885616215367923160215833549, −9.8432211069904777762181736527, −8.43029652237378053306939624808, −7.599772377495099895442958728183, −6.1903154696886167142918974753, −4.39723089149504165540525961283, −3.65433833315580173855447896643, −2.403580703045412306416982067265, 0.626361087623804875207223183351, 2.77188155243427776977590123938, 3.50439461787562276723558009822, 5.28546940636194900924333357675, 6.76476459310776566716940196102, 7.77740529878220026149801442081, 8.61942953328539473507805170412, 9.812437669231366308575496189522, 11.297999648041612062476625993010, 12.566030888527793860459231133725, 13.056221875027247250326603185114, 14.277279803078418024140619777047, 15.6915696115721110445214374345, 15.9985100227189849317990074808, 17.78435165647672212665588498392, 18.683983666601514795175393143741, 19.6470673186762518296877688563, 20.1330005190943006208020571762, 21.38015550692513988735693877665, 22.84800961674296840858316101423, 23.52126242302321338079017638496, 24.47992961497297460243108579433, 25.47150113781311244767637601464, 26.25271730454072175595491970551, 27.270406732478636252188858931136

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