↑

Properties

Label 1-311-311.61-r1-0-0
Degree $1$
Conductor $311$
Sign $0.881 + 0.471i$
Analytic cond. $33.4215$
Root an. cond. $33.4215$
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  + (−0.250 + 0.968i)2-s + (−0.250 − 0.968i)3-s + (−0.874 − 0.485i)4-s + (−0.440 − 0.897i)5-s + 6-s + (−0.994 + 0.101i)7-s + (0.688 − 0.724i)8-s + (−0.874 + 0.485i)9-s + (0.979 − 0.201i)10-s + (0.612 + 0.790i)11-s + (−0.250 + 0.968i)12-s + (−0.874 + 0.485i)13-s + (0.151 − 0.988i)14-s + (−0.758 + 0.651i)15-s + (0.528 + 0.848i)16-s + (−0.979 + 0.201i)17-s + ⋯
L(s)  = 1  + (−0.250 + 0.968i)2-s + (−0.250 − 0.968i)3-s + (−0.874 − 0.485i)4-s + (−0.440 − 0.897i)5-s + 6-s + (−0.994 + 0.101i)7-s + (0.688 − 0.724i)8-s + (−0.874 + 0.485i)9-s + (0.979 − 0.201i)10-s + (0.612 + 0.790i)11-s + (−0.250 + 0.968i)12-s + (−0.874 + 0.485i)13-s + (0.151 − 0.988i)14-s + (−0.758 + 0.651i)15-s + (0.528 + 0.848i)16-s + (−0.979 + 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5547223489 + 0.1389788382i\)
\(L(\frac12)\) \(\approx\) \(0.5547223489 + 0.1389788382i\)
\(L(1)\) \(\approx\) \(0.5619523681 + 0.02784772282i\)
\(L(1)\) \(\approx\) \(0.5619523681 + 0.02784772282i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.250 + 0.968i)T \)
3 \( 1 + (-0.250 - 0.968i)T \)
5 \( 1 + (-0.440 - 0.897i)T \)
7 \( 1 + (-0.994 + 0.101i)T \)
11 \( 1 + (0.612 + 0.790i)T \)
13 \( 1 + (-0.874 + 0.485i)T \)
17 \( 1 + (-0.979 + 0.201i)T \)
19 \( 1 + (0.440 - 0.897i)T \)
23 \( 1 + (-0.688 + 0.724i)T \)
29 \( 1 + (-0.151 - 0.988i)T \)
31 \( 1 + (-0.979 - 0.201i)T \)
37 \( 1 + (0.994 + 0.101i)T \)
41 \( 1 + (-0.820 - 0.571i)T \)
43 \( 1 + (0.994 - 0.101i)T \)
47 \( 1 + (0.151 + 0.988i)T \)
53 \( 1 + (-0.994 + 0.101i)T \)
59 \( 1 + (0.994 - 0.101i)T \)
61 \( 1 + (0.440 - 0.897i)T \)
67 \( 1 + (0.820 - 0.571i)T \)
71 \( 1 + (0.954 + 0.299i)T \)
73 \( 1 + (-0.0506 - 0.998i)T \)
79 \( 1 + (0.820 + 0.571i)T \)
83 \( 1 + (-0.758 - 0.651i)T \)
89 \( 1 + (-0.994 - 0.101i)T \)
97 \( 1 + (0.954 + 0.299i)T \)
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

−25.31469818270051867373604997479, −23.66414694283056522138499481227, −22.42962839402621460039822263686, −22.381836664456253211055137811748, −21.64172241168542449464666833989, −20.140858239306007037209772078235, −19.86213822258469376551832866173, −18.77265319369917555528319739529, −17.897782525231069389923454059763, −16.71557456656660206598708095025, −16.07170818858602190746841290431, −14.76799202480247369075118272048, −14.04441797984709329004634669409, −12.67679318462284261820790237283, −11.723218886625639884002317544821, −10.9032096699866218899771384433, −10.13744500500088430941018432332, −9.39109845267812695469828093852, −8.2954518095808492043974890543, −6.86348749009160713145190267747, −5.60695027333230634761714965425, −4.14838151928797298330704224911, −3.44495317023323926418451108610, −2.589096121495159532623155888353, −0.37816420742066771904263011232, 0.55529160463469617552754930662, 2.02310267838372019732365029783, 4.035264277506248933626727209202, 5.10067555938612469016472102960, 6.23587537051330525790472387740, 7.073671981831758286289496556262, 7.82711881147793326179906156244, 9.10201924296493170119580386300, 9.59629681276005756303786255532, 11.415554924924428467481824185983, 12.47466057616779261485741432279, 13.09119269189545805763707439617, 14.027396419346263275151953786224, 15.28780419197539594531862070158, 16.08160074908081641063545283500, 17.11505879602309515034360297088, 17.501994677455395282634996212430, 18.74883124971498306341195177610, 19.67034093775480639004857354585, 19.96669466139220441527296358526, 22.0867806892057204107946975246, 22.56870714212548879497418015386, 23.75581988790980691862029336067, 24.104220948740532875567916084600, 25.041979635351959541311051551771

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