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Properties

Label 1-405-405.158-r0-0-0
Degree $1$
Conductor $405$
Sign $-0.986 - 0.163i$
Analytic cond. $1.88081$
Root an. cond. $1.88081$
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  + (−0.230 + 0.973i)2-s + (−0.893 − 0.448i)4-s + (−0.549 + 0.835i)7-s + (0.642 − 0.766i)8-s + (0.993 − 0.116i)11-s + (−0.957 − 0.286i)13-s + (−0.686 − 0.727i)14-s + (0.597 + 0.802i)16-s + (−0.984 + 0.173i)17-s + (−0.173 + 0.984i)19-s + (−0.116 + 0.993i)22-s + (0.549 + 0.835i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.686 + 0.727i)29-s + ⋯
L(s)  = 1  + (−0.230 + 0.973i)2-s + (−0.893 − 0.448i)4-s + (−0.549 + 0.835i)7-s + (0.642 − 0.766i)8-s + (0.993 − 0.116i)11-s + (−0.957 − 0.286i)13-s + (−0.686 − 0.727i)14-s + (0.597 + 0.802i)16-s + (−0.984 + 0.173i)17-s + (−0.173 + 0.984i)19-s + (−0.116 + 0.993i)22-s + (0.549 + 0.835i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.686 + 0.727i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.986 - 0.163i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ -0.986 - 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04383417740 + 0.5335282218i\)
\(L(\frac12)\) \(\approx\) \(-0.04383417740 + 0.5335282218i\)
\(L(1)\) \(\approx\) \(0.5451125401 + 0.4242116030i\)
\(L(1)\) \(\approx\) \(0.5451125401 + 0.4242116030i\)

Euler product

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

−23.74998328565017704394814027773, −22.566643586057617761596804345431, −22.29912828553760823768186505964, −21.21211563607057761396412868201, −20.18742142591784903406756162183, −19.62862196859546422998864770471, −19.015854997114033019657618433379, −17.64568695811670644251515374028, −17.19047682244612219461519146158, −16.25052549610898048561864299695, −14.82167025959007746609104950241, −13.91260376249391468728471538817, −13.08569908627012780352530574183, −12.229514994179215587428337455508, −11.25391642190759637159173090749, −10.45542447042391608581201770404, −9.4320796776861379009302157178, −8.85277371410103173097433333348, −7.38824715158070335831958162436, −6.60560283126827749271510370673, −4.83902150861707606682013642645, −4.11594967225001022245900667358, −2.98698323318379478180983989705, −1.82880626602812198541454311107, −0.33750624636604231368768314620, 1.67161273325919383592290016728, 3.300797733923678295525817770512, 4.52963300852519466899270958877, 5.63443555791569209291490560810, 6.42468518266946433574745848376, 7.35808938741979978436633873451, 8.50102532739456842893373128262, 9.298711367180359409673547146, 10.006155923106608768786552119705, 11.43057889154878006562608658195, 12.53839505473982443952691499298, 13.345337801917478280364211450, 14.571015698714235587848190832597, 15.080144437625981951767858736959, 16.01576477330899542248248250727, 16.94452279982431085552434617873, 17.572078716604525499166752655920, 18.736259667081375049714904483811, 19.26400414349735837880176403505, 20.24992766466973381566279240560, 21.87420445664657817513782777173, 22.1916979033535516163284967448, 23.08087044197825372845629436410, 24.24732842451771619514385971012, 24.80488110050176450603731768864

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