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Properties

Label 1-1620-1620.23-r1-0-0
Degree $1$
Conductor $1620$
Sign $0.105 - 0.994i$
Analytic cond. $174.093$
Root an. cond. $174.093$
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  + (−0.998 − 0.0581i)7-s + (0.597 + 0.802i)11-s + (0.727 − 0.686i)13-s + (−0.984 + 0.173i)17-s + (0.173 − 0.984i)19-s + (0.998 − 0.0581i)23-s + (0.973 + 0.230i)29-s + (−0.893 − 0.448i)31-s + (−0.342 + 0.939i)37-s + (0.286 − 0.957i)41-s + (0.116 + 0.993i)43-s + (−0.448 − 0.893i)47-s + (0.993 + 0.116i)49-s + (−0.866 + 0.5i)53-s + (−0.597 + 0.802i)59-s + ⋯
L(s)  = 1  + (−0.998 − 0.0581i)7-s + (0.597 + 0.802i)11-s + (0.727 − 0.686i)13-s + (−0.984 + 0.173i)17-s + (0.173 − 0.984i)19-s + (0.998 − 0.0581i)23-s + (0.973 + 0.230i)29-s + (−0.893 − 0.448i)31-s + (−0.342 + 0.939i)37-s + (0.286 − 0.957i)41-s + (0.116 + 0.993i)43-s + (−0.448 − 0.893i)47-s + (0.993 + 0.116i)49-s + (−0.866 + 0.5i)53-s + (−0.597 + 0.802i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.105 - 0.994i$
Analytic conductor: \(174.093\)
Root analytic conductor: \(174.093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1620,\ (1:\ ),\ 0.105 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.015625454 - 0.9135059470i\)
\(L(\frac12)\) \(\approx\) \(1.015625454 - 0.9135059470i\)
\(L(1)\) \(\approx\) \(0.9559666491 - 0.07019122868i\)
\(L(1)\) \(\approx\) \(0.9559666491 - 0.07019122868i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.998 - 0.0581i)T \)
11 \( 1 + (0.597 + 0.802i)T \)
13 \( 1 + (0.727 - 0.686i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.998 - 0.0581i)T \)
29 \( 1 + (0.973 + 0.230i)T \)
31 \( 1 + (-0.893 - 0.448i)T \)
37 \( 1 + (-0.342 + 0.939i)T \)
41 \( 1 + (0.286 - 0.957i)T \)
43 \( 1 + (0.116 + 0.993i)T \)
47 \( 1 + (-0.448 - 0.893i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.597 + 0.802i)T \)
61 \( 1 + (-0.835 - 0.549i)T \)
67 \( 1 + (0.230 + 0.973i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.642 + 0.766i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (0.957 - 0.286i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.918 - 0.396i)T \)
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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

−20.370380535064945385016343159945, −19.545639597545972799729245885558, −19.04363660923052831228641570366, −18.34038250491119396248744860558, −17.416590733954593831815223946519, −16.48164986559125574100780805949, −16.14834335138920719491675691335, −15.33380848154139015569387784068, −14.25411102307135960625142905624, −13.74589242074610273142514771298, −12.90218567859589975088682211076, −12.19909712547077678860170259202, −11.23024173907643446885101093628, −10.70731877395555877023110512409, −9.54454883622485902456083360224, −9.04967329438805936504147043156, −8.29401963419182194187528974195, −7.098356057510603522844563963703, −6.421328660983636252180543715949, −5.84550891363016758727714976785, −4.6511426233891252514980455773, −3.6845700120916065796794586158, −3.11102082252564452491860136501, −1.88471702793401310432192904552, −0.836935907944681253607134195981, 0.31607675349128347416613606061, 1.375812593468618261408201078292, 2.59111822778896030982764293424, 3.3560666976423159411524009363, 4.30030928743229965388282269271, 5.149948947797236730997764015262, 6.31127499587171371098125042332, 6.75274332600989284066149061684, 7.626685894164876204258739188115, 8.84950997505308646435742371687, 9.21661978822041511577752475855, 10.21851060173182586587815577069, 10.90060007714988137120687253273, 11.775786001210275203166101896359, 12.759176180053712729465619928, 13.14911152864536519352335358214, 13.9725496355142922616936044458, 15.11097715886454429938779221841, 15.49762727298129802216958717171, 16.30625021259990813984774574095, 17.21147029186239456118329260257, 17.77310953519245017349393157390, 18.61141020711525131618998645016, 19.5084662963138726870261403439, 20.00961488266596330521962708154

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