↑

Properties

Label 1-344-344.251-r1-0-0
Degree $1$
Conductor $344$
Sign $0.736 + 0.675i$
Analytic cond. $36.9679$
Root an. cond. $36.9679$
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  + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.736 + 0.675i$
Analytic conductor: \(36.9679\)
Root analytic conductor: \(36.9679\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (1:\ ),\ 0.736 + 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.923069612 + 0.7484146611i\)
\(L(\frac12)\) \(\approx\) \(1.923069612 + 0.7484146611i\)
\(L(1)\) \(\approx\) \(1.145573132 + 0.2545904551i\)
\(L(1)\) \(\approx\) \(1.145573132 + 0.2545904551i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + T \)
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

−24.58723361651470207171046947224, −23.54415292660356063349812707181, −22.94293062802125493278047362300, −22.08246462373993856367797344879, −21.19002678825889833597950427584, −19.78421749303643134070023357897, −19.34082749684016905155873687114, −18.0679156850644187347692471142, −17.476899126923622206474601014589, −17.01892709663003807530211390421, −15.45036384157530855208292418535, −14.44761142532221476480976649837, −13.59676910861186704383165913901, −12.96138001834704566730058914696, −11.53829297430774876341013180308, −10.97031897176467110096539233130, −10.10295272949097196067223861290, −8.597438849872520857534102402399, −7.54184415260299451856217867601, −6.64837589363060443086051450479, −6.004409378901048875945556432723, −4.6128160539644816074662531483, −3.22398019897840068661673390782, −1.867812775320771913766830504244, −0.85020341148083560926191489048, 0.9651224271423183922897313964, 2.30913799522612325555108184618, 4.026816695087958746901176300, 4.76598770497180351054652124410, 5.77106088348122346025127242654, 6.59372059497920970100331532031, 8.540857913158863338327265125, 9.02275961753665243735263917989, 9.867542589509011638335272718, 11.20561436033817633212140416161, 11.84617909235439774400335898302, 12.76714733822815119402415799679, 14.175718503168388398137561294966, 14.82025480418903467297472256051, 16.12276587167021251988993558030, 16.52410485681070029130780222903, 17.53403133449261488327399227356, 18.30661270473481096811226055698, 19.58044577318884445914926747189, 20.82378865928046237587865409882, 21.087172561700069396995107954, 22.051133170461433448598473780382, 22.81676615575215117434876523042, 23.98116375675602234013756354889, 24.79503170648393366283778498396

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