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Properties

Label 1-1224-1224.965-r1-0-0
Degree $1$
Conductor $1224$
Sign $-0.208 + 0.978i$
Analytic cond. $131.537$
Root an. cond. $131.537$
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  + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + 19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.866 − 0.5i)31-s + 35-s − i·37-s + (−0.866 − 0.5i)41-s + (−0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + 19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.866 − 0.5i)31-s + 35-s − i·37-s + (−0.866 − 0.5i)41-s + (−0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.208 + 0.978i$
Analytic conductor: \(131.537\)
Root analytic conductor: \(131.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1224,\ (1:\ ),\ -0.208 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2016491130 + 0.2490964513i\)
\(L(\frac12)\) \(\approx\) \(0.2016491130 + 0.2490964513i\)
\(L(1)\) \(\approx\) \(0.7002892935 - 0.07055187723i\)
\(L(1)\) \(\approx\) \(0.7002892935 - 0.07055187723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - iT \)
73 \( 1 + iT \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−20.51089955402712041730724656339, −19.95073131006436873962808275921, −19.15876155856817111656511371964, −18.54542588479329777530903893603, −17.876844612852436899923663026746, −16.53309984760072294927869115079, −16.11833014902767625787785848216, −15.57882820297550796617328360882, −14.4710749064388093141121297167, −13.72643169892479796753652869035, −13.06875451656927508503498275905, −11.97464353764213974796700587541, −11.41034807910877023235265420310, −10.47424578760860358026354408074, −9.86871442941331269803873403685, −8.74239138970746256482517970744, −7.93807622499077391845229477982, −7.08645606920487579540827932866, −6.46555631684338435258475024416, −5.3876732469686205928561056889, −4.26292244520552652779558047958, −3.427501373203111316746100058143, −2.85011587896978667757396020737, −1.32958168168276996625199816275, −0.09757910383856632112209313039, 0.68782903526556603564004102941, 2.18173561029707173664899552231, 3.19255976014633640180561919250, 3.9261934090498171703055307241, 5.085906218953016691582540738363, 5.71944459124240391933001910141, 6.83424033288423088158434944762, 7.77911176138289681903010055089, 8.342832921136823913776016075870, 9.337243863996589914113869068298, 10.10771742342900755874308966753, 10.99123200712170624182808664130, 11.99822866817527881743645839873, 12.58463311037827439959326528566, 13.158674274548792278398481421362, 14.1868028980993486535644408789, 15.399759365848347280518178511009, 15.713716859412608788152328195472, 16.26279192282640975168079176034, 17.33581059263087960866444618115, 18.294281318558343297964025454617, 18.77639572251039091499308329557, 19.79915105933594998861415180868, 20.290103983213354960377193128557, 20.95133907239934726080036665130

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