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Properties

Label 1-344-344.123-r0-0-0
Degree $1$
Conductor $344$
Sign $0.0861 + 0.996i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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  + (0.5 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0861 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0861 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.0861 + 0.996i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 0.0861 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9342438307 + 0.8569679711i\)
\(L(\frac12)\) \(\approx\) \(0.9342438307 + 0.8569679711i\)
\(L(1)\) \(\approx\) \(1.037824855 + 0.3947995588i\)
\(L(1)\) \(\approx\) \(1.037824855 + 0.3947995588i\)

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 \( 1 + T \)
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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

−24.54780791271210934912958688699, −23.91565675641763797401456911816, −22.75172742193526042403141074587, −22.53428068028371293119339707841, −20.91956583718621385518980756976, −19.99926465938060833009641407288, −19.32609519238036516471319927935, −18.669124723535035287978639061, −17.69716089363999298954797445451, −16.704528507002833673000722683798, −15.58408895291233099855582953966, −14.51715343583514379313184871455, −13.84214073453023808324303990835, −13.10793326835084462132749944411, −11.69495172146654569274435257950, −11.28145826486949931436224970014, −9.79878345050977438602318356603, −8.89259356701774117637488121854, −7.66628331960388158273171268626, −6.7748536621614979061725869672, −6.45668023727984417768536344593, −4.327418397959569597871091225478, −3.417966770713466605220784469074, −2.35962482748163942178950635823, −0.79782100714093808084000850082, 1.570916559517032991313955171728, 3.22709242053156097184739667087, 3.902062443163436764394017925, 5.151216504103358571133637938190, 6.00028295996432403836751121088, 7.665490050806906310352216205093, 8.77271769566104545415001665637, 9.089291962219834581485723007747, 10.27471695320333357840942378872, 11.40741228533764619948645120223, 12.39216895156503573349177626171, 13.24413576209867387297408160711, 14.50811702241262402101583145006, 15.35244974830268925374694932397, 16.00597585069072443454323853298, 16.77725028839771446561330626600, 17.88752370916114019155537680551, 19.3920689094566877336285128348, 19.66597127491902846610250263254, 20.71007853405770779764955372832, 21.480901116006424906193792555868, 22.39922278161420126152025943543, 23.12454891469749669995049563631, 24.56274633954021892127389398236, 25.058284807424420489646359677607

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