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Properties

Label 1-465-465.74-r0-0-0
Degree $1$
Conductor $465$
Sign $-0.984 - 0.175i$
Analytic cond. $2.15945$
Root an. cond. $2.15945$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.104 + 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.913 + 0.406i)11-s + (−0.978 + 0.207i)13-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.913 + 0.406i)17-s + (−0.978 − 0.207i)19-s + (−0.104 + 0.994i)22-s + (0.809 + 0.587i)23-s + (−0.5 − 0.866i)26-s + (−0.669 − 0.743i)28-s + (0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.104 + 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.913 + 0.406i)11-s + (−0.978 + 0.207i)13-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.913 + 0.406i)17-s + (−0.978 − 0.207i)19-s + (−0.104 + 0.994i)22-s + (0.809 + 0.587i)23-s + (−0.5 − 0.866i)26-s + (−0.669 − 0.743i)28-s + (0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.984 - 0.175i$
Analytic conductor: \(2.15945\)
Root analytic conductor: \(2.15945\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 465,\ (0:\ ),\ -0.984 - 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08637338532 + 0.9741602030i\)
\(L(\frac12)\) \(\approx\) \(-0.08637338532 + 0.9741602030i\)
\(L(1)\) \(\approx\) \(0.6770066188 + 0.7057684394i\)
\(L(1)\) \(\approx\) \(0.6770066188 + 0.7057684394i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.104 + 0.994i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (-0.978 - 0.207i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 - T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−23.13814296163464696524573131472, −22.58286235752318498921824041471, −21.68836058842911326499469940239, −20.87165287627072667054039343831, −19.92785804333935957495912207752, −19.50250970248308874265596817951, −18.53678168288251706390103249334, −17.28618082861893847280869057652, −16.984532956299781446759077792980, −15.40131429444821712113490658907, −14.45377047202156029186501071825, −13.8040178497146235067579007129, −12.89832767044134223257761862203, −12.00797808720076383944749348456, −11.05451751260601253766439063680, −10.39308028775786797243055770891, −9.38810941028135742317970468024, −8.50737050089436951139704361647, −7.15949982274323958869546836770, −6.14110123571356516478232270724, −4.738125267808067515113187322480, −4.150144136053348265081698315312, −2.98749892132787133876228148677, −1.82163876005873738274337967690, −0.4804676340612124624599274013, 1.918739483873359824416757426056, 3.25263455437396465812389351311, 4.53011740811644325915678421692, 5.2082795708847607300704421003, 6.47609183121444324063264454104, 6.9801957049369218468719670212, 8.36922740859107605387372736916, 8.963986936502735183210828720287, 9.879732592619579394403708301311, 11.43080651758439745389665013338, 12.29929328272520644101585909815, 13.02231854946133889281184199829, 14.156513016434336754034847049494, 15.11697727057259663853615947314, 15.29649351332895223167256884036, 16.713978096749784047710831678594, 17.25906279512412719718093852361, 18.134181101906714358162557833664, 19.09776586336659069948976752028, 19.963503873092166521704801771469, 21.4566773003851642538622574011, 21.84952573813101245003714167505, 22.62230173257007984465175673018, 23.64987016542852041349644879686, 24.41148789138060738465015070488

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