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Properties

Label 1-468-468.295-r1-0-0
Degree $1$
Conductor $468$
Sign $0.996 + 0.0807i$
Analytic cond. $50.2935$
Root an. cond. $50.2935$
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  + (−0.5 + 0.866i)5-s − 7-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.5 − 0.866i)35-s + (−0.5 − 0.866i)37-s + 41-s − 43-s + (0.5 + 0.866i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s − 7-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.5 − 0.866i)35-s + (−0.5 − 0.866i)37-s + 41-s − 43-s + (0.5 + 0.866i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.996 + 0.0807i$
Analytic conductor: \(50.2935\)
Root analytic conductor: \(50.2935\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 468,\ (1:\ ),\ 0.996 + 0.0807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.270611090 + 0.05139539412i\)
\(L(\frac12)\) \(\approx\) \(1.270611090 + 0.05139539412i\)
\(L(1)\) \(\approx\) \(0.8684744642 + 0.07037359885i\)
\(L(1)\) \(\approx\) \(0.8684744642 + 0.07037359885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 - T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−23.5228044874960109223587523801, −22.80573995253330650689105106760, −22.17294791418603600433376530497, −20.86673106787441800359673096066, −20.16450413164897102254252030075, −19.5835467562051546036482289134, −18.59476422371914580248544596190, −17.55624220031618708119975204552, −16.6044043282723266933458856975, −15.96505018840357782601229573641, −15.209022336950574427209665704876, −13.95496784543246855415591728661, −13.10477507456313627224627837645, −12.16082474651815920705847171844, −11.713037919637006630721010776109, −10.11677962371534377685996452940, −9.514711544661627228661548537787, −8.54629685347190928559943763642, −7.482366004640878160252159757826, −6.561031999426042010992485032493, −5.397417677438080574318859562, −4.3259887001731795694132341824, −3.47645747867664351202943181300, −2.02633419989479749594060503007, −0.64540909289734306680930853941, 0.56025994344595041070535702711, 2.350125906913682785332306928, 3.40090599895348654843692509379, 4.08996443575578192143701065174, 5.78035837889664774526000086164, 6.51902234826495975810918572194, 7.38135145586178667417097809865, 8.531617962803110231916813978935, 9.49616997481390137206487997492, 10.53562792123008656451631502593, 11.28524677089255265741108396633, 12.22196482696330269917168702160, 13.28852423936666995086986601910, 14.0954831984658265565526272038, 15.09823584238685607819824786768, 15.87475152675398531991385782171, 16.63626372073288745341385265648, 17.78450984652532931921388557727, 18.640089556839905429149957572467, 19.569957614287508992180392462292, 19.80710779320238391149655294421, 21.330372335704777329631036184112, 22.3078185174501549883698532949, 22.45861570465071904573204948511, 23.74398337286135644741638504222

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