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Properties

Label 1-405-405.23-r0-0-0
Degree $1$
Conductor $405$
Sign $0.634 - 0.772i$
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  + (0.957 − 0.286i)2-s + (0.835 − 0.549i)4-s + (0.998 + 0.0581i)7-s + (0.642 − 0.766i)8-s + (−0.597 − 0.802i)11-s + (0.727 − 0.686i)13-s + (0.973 − 0.230i)14-s + (0.396 − 0.918i)16-s + (−0.984 + 0.173i)17-s + (−0.173 + 0.984i)19-s + (−0.802 − 0.597i)22-s + (−0.998 + 0.0581i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (0.973 + 0.230i)29-s + ⋯
L(s)  = 1  + (0.957 − 0.286i)2-s + (0.835 − 0.549i)4-s + (0.998 + 0.0581i)7-s + (0.642 − 0.766i)8-s + (−0.597 − 0.802i)11-s + (0.727 − 0.686i)13-s + (0.973 − 0.230i)14-s + (0.396 − 0.918i)16-s + (−0.984 + 0.173i)17-s + (−0.173 + 0.984i)19-s + (−0.802 − 0.597i)22-s + (−0.998 + 0.0581i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (0.973 + 0.230i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.634 - 0.772i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ 0.634 - 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.337725715 - 1.105196708i\)
\(L(\frac12)\) \(\approx\) \(2.337725715 - 1.105196708i\)
\(L(1)\) \(\approx\) \(1.892420624 - 0.5385268888i\)
\(L(1)\) \(\approx\) \(1.892420624 - 0.5385268888i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.957 - 0.286i)T \)
7 \( 1 + (0.998 + 0.0581i)T \)
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

−24.263916886208137875411208340101, −23.62608651309520629180695352605, −22.88682098488787405186062539310, −21.80911688895896454977831649183, −21.138531668086787521264083518127, −20.412996473601692800714770110584, −19.53896305829708427767757884636, −17.98313749512023425877220351194, −17.58538426973286169774858648113, −16.269581510552192052166671143582, −15.53949749356875692094357323469, −14.74511131316014156628885177505, −13.78550596917540619407485146470, −13.1577755026303753333453402677, −11.9458439343488342225898033586, −11.30653907812858766589569129248, −10.35398034695002890025993776260, −8.80905037414399732342537893983, −7.87991628711973497244601560320, −6.93247677765795504851038552911, −5.96659525516613939518402632820, −4.64968493524427370247639037066, −4.33468972190392482837759958614, −2.67995422807513089161626106553, −1.76852464931835837533329477088, 1.28114207420602984102517944156, 2.45272899141686074596905763825, 3.60278619065756833735315966773, 4.63601469426550518565466656664, 5.60937968029710346576651037813, 6.410093886218395697947635724986, 7.83783387258799760920636597164, 8.56231269264094050370115211860, 10.303214942098773422176871700076, 10.82733020327060407659151534623, 11.77341465480197154955109823987, 12.65963356919153886481067166230, 13.7622446242161250952810413803, 14.17853895887343276509742939745, 15.48739070102809817933275180074, 15.85035552923617776527338336095, 17.21171548602537840791453806045, 18.23198642927646355234299389919, 19.09517883355595391273403358232, 20.25813639608489805146702895548, 20.819291867530889630479989357693, 21.61086495001127306844014702754, 22.41439133749498055989606076969, 23.44846910989566808858802590064, 24.03221123715182110256144820801

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