Properties

Label 1-20e2-400.123-r0-0-0
Degree $1$
Conductor $400$
Sign $0.780 + 0.625i$
Analytic cond. $1.85759$
Root an. cond. $1.85759$
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  + (0.809 + 0.587i)3-s i·7-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.309 + 0.951i)13-s + (0.587 + 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (0.951 + 0.309i)23-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)33-s + (0.309 + 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s i·7-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.309 + 0.951i)13-s + (0.587 + 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (0.951 + 0.309i)23-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)33-s + (0.309 + 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.780 + 0.625i$
Analytic conductor: \(1.85759\)
Root analytic conductor: \(1.85759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (0:\ ),\ 0.780 + 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.635194932 + 0.5742395879i\)
\(L(\frac12)\) \(\approx\) \(1.635194932 + 0.5742395879i\)
\(L(1)\) \(\approx\) \(1.355397784 + 0.2640152612i\)
\(L(1)\) \(\approx\) \(1.355397784 + 0.2640152612i\)

Euler product

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

−24.54637530206502110495869059391, −23.45597879370588163958002176146, −22.70830817917925611876710941916, −21.454757874418473187031437304086, −20.78317239120092963203074487492, −19.94072944120668724027463895057, −18.999192940388084219235927622059, −18.17590617013334112775768668610, −17.75899591871135979262531481384, −16.04450723347830803745966330637, −15.42416086310920424121721510957, −14.58940507638872771078068553172, −13.547329052873907760851423120567, −12.744258060956240179261796052592, −12.057483848590720503622906034129, −10.77879361917582143623316983845, −9.592080952462336079543578268576, −8.77471424674138586793016076076, −7.882400321125824277839269323633, −7.04225705512778242732026245535, −5.76875728237226410133268468261, −4.812651460234101181123047387594, −2.95554199635257844673464572283, −2.733809592859278877465987826218, −1.10089616190565854819133761504, 1.41139430107975892647173847220, 2.82674858175727573275162463949, 3.814214076984006879840374304755, 4.645814802486849200536089654789, 5.93616403326962223527416141006, 7.36205127852082212477246875607, 8.03586756143037178322842884782, 9.07127625096497931122391358603, 10.17098500473771160909346548523, 10.63820625967885888817132625247, 11.90075800261550112323969268354, 13.36214376395878045057729817161, 13.72685620175295976873348159769, 14.7250416718062135917689787097, 15.661098644036351798585968245486, 16.50711237098117579005805241780, 17.22043980824617392130445769951, 18.73703998023324666070635821870, 19.18972368164098586792634758801, 20.34000971381034561401146559727, 20.96606330446608823554473851932, 21.551525778273594242383683862181, 22.825516187516026789404073205925, 23.59217953425487196589914652836, 24.48178462181982390860273797104

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