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Properties

Label 1-341-341.335-r0-0-0
Degree $1$
Conductor $341$
Sign $-0.145 + 0.989i$
Analytic cond. $1.58359$
Root an. cond. $1.58359$
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  + (−0.809 + 0.587i)2-s + (−0.978 − 0.207i)3-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (0.913 − 0.406i)6-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.913 − 0.406i)17-s + (−0.978 + 0.207i)18-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.978 − 0.207i)3-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (0.913 − 0.406i)6-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.913 − 0.406i)17-s + (−0.978 + 0.207i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(341\)    =    \(11 \cdot 31\)
Sign: $-0.145 + 0.989i$
Analytic conductor: \(1.58359\)
Root analytic conductor: \(1.58359\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{341} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 341,\ (0:\ ),\ -0.145 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4480391712 + 0.5187006902i\)
\(L(\frac12)\) \(\approx\) \(0.4480391712 + 0.5187006902i\)
\(L(1)\) \(\approx\) \(0.5676161462 + 0.2880025594i\)
\(L(1)\) \(\approx\) \(0.5676161462 + 0.2880025594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
7 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + T \)
29 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−24.74305132445253497044716302108, −23.5669801074990029836865004372, −23.14074199912663162682483304634, −21.61834387747056216003300971545, −21.057696546526740727133351905863, −20.404467236908453936293799098492, −19.34615628715843892792839224093, −18.25106968686686128383562725135, −17.48633420964987429799351134698, −16.717538051626924759104312854223, −16.28650395530828655020296976764, −15.02135222611720272483103425045, −13.292287027828210618722236127119, −12.63033872701099013035125070715, −11.68965441961540401235407620940, −10.81431653169049442889150923339, −10.20026013650870120120065750544, −8.93823360218740816561269306077, −8.08937571566212028540024067864, −7.03029134745888146507616257157, −5.67234540313060668442536981778, −4.49588387571685999986298901950, −3.645921272833300495278339861133, −1.568206409687874415544218145423, −0.794962297422011781204554731794, 1.25069875703348975417773022475, 2.5283642784957545470388120564, 4.49897659991135799647248702926, 5.72786733341127313162338483062, 6.35632788809633843249693576998, 7.331300201745284899926858192917, 8.256786139273607032421670499732, 9.4849777996704374551963401263, 10.62082869796181749252088707615, 11.20879334565683172245943049736, 11.9821553103725169364281096698, 13.539651768831932152916466417185, 14.68595007117500812899965913198, 15.38173311720668190613672444925, 16.300050378383654489882700047345, 17.29132457692961581572318112062, 18.01512650216897984399238919568, 18.788196469829202863869535658156, 19.13430644193652012180934687192, 20.8971725315119788453418107861, 21.654904981765063981438635141782, 22.96194820369866092575521637574, 23.30948996205607522280544052940, 24.32666679640602414537451676774, 25.21923545481552729977444273571

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