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Properties

Label 1-405-405.148-r1-0-0
Degree $1$
Conductor $405$
Sign $-0.490 + 0.871i$
Analytic cond. $43.5232$
Root an. cond. $43.5232$
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  + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.835 − 0.549i)11-s + (0.116 − 0.993i)13-s + (−0.597 + 0.802i)14-s + (−0.0581 + 0.998i)16-s + (−0.642 − 0.766i)17-s + (−0.766 − 0.642i)19-s + (0.549 + 0.835i)22-s + (0.230 + 0.973i)23-s + (−0.5 + 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.597 − 0.802i)29-s + ⋯
L(s)  = 1  + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.835 − 0.549i)11-s + (0.116 − 0.993i)13-s + (−0.597 + 0.802i)14-s + (−0.0581 + 0.998i)16-s + (−0.642 − 0.766i)17-s + (−0.766 − 0.642i)19-s + (0.549 + 0.835i)22-s + (0.230 + 0.973i)23-s + (−0.5 + 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.597 − 0.802i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.490 + 0.871i$
Analytic conductor: \(43.5232\)
Root analytic conductor: \(43.5232\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (1:\ ),\ -0.490 + 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06930522245 - 0.1185865562i\)
\(L(\frac12)\) \(\approx\) \(-0.06930522245 - 0.1185865562i\)
\(L(1)\) \(\approx\) \(0.5359806133 - 0.2243017398i\)
\(L(1)\) \(\approx\) \(0.5359806133 - 0.2243017398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.918 - 0.396i)T \)
7 \( 1 + (0.230 - 0.973i)T \)
11 \( 1 + (-0.835 - 0.549i)T \)
13 \( 1 + (0.116 - 0.993i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.230 + 0.973i)T \)
29 \( 1 + (-0.597 - 0.802i)T \)
31 \( 1 + (-0.286 + 0.957i)T \)
37 \( 1 + (0.984 - 0.173i)T \)
41 \( 1 + (0.396 + 0.918i)T \)
43 \( 1 + (0.448 + 0.893i)T \)
47 \( 1 + (-0.957 + 0.286i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.835 - 0.549i)T \)
61 \( 1 + (-0.686 + 0.727i)T \)
67 \( 1 + (-0.802 - 0.597i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.342 - 0.939i)T \)
79 \( 1 + (-0.396 + 0.918i)T \)
83 \( 1 + (0.918 + 0.396i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.998 + 0.0581i)T \)
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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

−24.734830762352884407790830771151, −24.06351209975682946120103581113, −23.26666730334326776094572525988, −22.016153117910800065626966371256, −21.05228239507996475639322470333, −20.33866395498767985026021497758, −19.10204692259449993758505171614, −18.61948310371140654024392855156, −17.82451129683088305592071930104, −16.832862019712032792786530616882, −16.04015569045251653751388383236, −15.03352189102122191223591332747, −14.61059662574857998117779404046, −13.06814618993929078699251014259, −12.07936995436472836587430917942, −11.06031856618238038802857491192, −10.25194230076791623790352152382, −9.09561481829097412774649383698, −8.52096230977486058545475487877, −7.47720177346305807622267398758, −6.40715254235213466963011272394, −5.56205761193936535053268910156, −4.34474655301293708461074841341, −2.45322245845773125411715433728, −1.79111697659162914630577337601, 0.05596108957956790466864693234, 1.03136206126137478789981356739, 2.49611072505088546353564100728, 3.46488694007325160437877132443, 4.775348640252864484865426174432, 6.20843332856469335694456019570, 7.40097030656535824615365113564, 7.97497459612644936265762833585, 9.06692451593704286956053449427, 10.09942836853401088920666462997, 10.91376820002951390687602455310, 11.46538238839206640932826446687, 13.01713502892237984043956695031, 13.36712674152527561576033092587, 14.91877410541089758121604986252, 15.88588061871991013082028452025, 16.62788835885167413309963579106, 17.73385844121430591472462234957, 18.041102695151963679803583813548, 19.37362876835298105000022838967, 19.90902276844050445890780135435, 20.851112675470122767373845323508, 21.45141170146603046205072872120, 22.66440396088730810208619234346, 23.62235108137563508782640360764

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