Properties

Label 1-311-311.206-r1-0-0
Degree $1$
Conductor $311$
Sign $0.371 + 0.928i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.994 + 0.101i)2-s + (−0.994 − 0.101i)3-s + (0.979 − 0.201i)4-s + (0.688 + 0.724i)5-s + 6-s + (0.347 − 0.937i)7-s + (−0.954 + 0.299i)8-s + (0.979 + 0.201i)9-s + (−0.758 − 0.651i)10-s + (0.0506 + 0.998i)11-s + (−0.994 + 0.101i)12-s + (0.979 + 0.201i)13-s + (−0.250 + 0.968i)14-s + (−0.612 − 0.790i)15-s + (0.918 − 0.394i)16-s + (0.758 + 0.651i)17-s + ⋯
L(s)  = 1  + (−0.994 + 0.101i)2-s + (−0.994 − 0.101i)3-s + (0.979 − 0.201i)4-s + (0.688 + 0.724i)5-s + 6-s + (0.347 − 0.937i)7-s + (−0.954 + 0.299i)8-s + (0.979 + 0.201i)9-s + (−0.758 − 0.651i)10-s + (0.0506 + 0.998i)11-s + (−0.994 + 0.101i)12-s + (0.979 + 0.201i)13-s + (−0.250 + 0.968i)14-s + (−0.612 − 0.790i)15-s + (0.918 − 0.394i)16-s + (0.758 + 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.371 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9048540990 + 0.6122517454i\)
\(L(\frac12)\) \(\approx\) \(0.9048540990 + 0.6122517454i\)
\(L(1)\) \(\approx\) \(0.6856489059 + 0.1475121719i\)
\(L(1)\) \(\approx\) \(0.6856489059 + 0.1475121719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.994 + 0.101i)T \)
3 \( 1 + (-0.994 - 0.101i)T \)
5 \( 1 + (0.688 + 0.724i)T \)
7 \( 1 + (0.347 - 0.937i)T \)
11 \( 1 + (0.0506 + 0.998i)T \)
13 \( 1 + (0.979 + 0.201i)T \)
17 \( 1 + (0.758 + 0.651i)T \)
19 \( 1 + (-0.688 + 0.724i)T \)
23 \( 1 + (0.954 - 0.299i)T \)
29 \( 1 + (0.250 + 0.968i)T \)
31 \( 1 + (0.758 - 0.651i)T \)
37 \( 1 + (-0.347 - 0.937i)T \)
41 \( 1 + (-0.528 - 0.848i)T \)
43 \( 1 + (-0.347 + 0.937i)T \)
47 \( 1 + (-0.250 - 0.968i)T \)
53 \( 1 + (0.347 - 0.937i)T \)
59 \( 1 + (-0.347 + 0.937i)T \)
61 \( 1 + (-0.688 + 0.724i)T \)
67 \( 1 + (0.528 - 0.848i)T \)
71 \( 1 + (0.874 + 0.485i)T \)
73 \( 1 + (0.820 - 0.571i)T \)
79 \( 1 + (0.528 + 0.848i)T \)
83 \( 1 + (-0.612 + 0.790i)T \)
89 \( 1 + (0.347 + 0.937i)T \)
97 \( 1 + (0.874 + 0.485i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.91106051009643791919274780901, −24.21566226567802179651194333637, −23.21634465242036312765870901892, −21.69843909814459726265376864079, −21.346930352542665044630803294472, −20.559036090546870330825827817451, −19.01567526500630559624050198607, −18.49155832269946370191447901227, −17.507773576980350426264489002317, −16.93495765782660380209450065431, −15.991816030368337128985752560949, −15.352574281756787740414305251751, −13.62596227394073504074062019695, −12.53465822848824997716746600767, −11.647139875211704991151913242704, −10.93614319939657073018527048395, −9.843146428155982696323610133484, −8.91802214641144724625960682139, −8.16542158855448447263649646943, −6.543171858925352657045121368603, −5.85582449783903192534817642444, −4.919153466983595250563777648266, −3.01639448038303343184090965470, −1.51876132597866513190399591966, −0.63114450444946619071710485211, 1.09593720648970727789014245202, 1.938009513634569521271537607630, 3.763154165942292378462161115006, 5.32560235758411967417514665119, 6.467299506478768289034470391703, 6.983335360341502430044435116404, 8.072701236674001274022481375051, 9.56629578453340206357505422628, 10.579509966184468261213047190909, 10.70360787038104523469159910426, 11.97812704769027593636278194901, 13.08943175785547970379776947453, 14.42660706645037872627442618211, 15.301813542701793591253474316206, 16.65212648430057847418612890541, 17.05841909712155972094826641508, 17.960957028822274325818368308205, 18.5141239011620962236647306107, 19.53045255847156906337174496091, 20.93606970159771826839552501498, 21.25466444023285590590781136584, 22.87770659183374823101488856310, 23.26009992421294527647227954172, 24.36157561930511649028755696445, 25.45712741730670603527655096991

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