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Properties

Label 1-456-456.155-r1-0-0
Degree $1$
Conductor $456$
Sign $0.939 - 0.341i$
Analytic cond. $49.0040$
Root an. cond. $49.0040$
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  + (−0.939 − 0.342i)5-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.173 − 0.984i)17-s + (−0.939 + 0.342i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s + (−0.766 + 0.642i)35-s + 37-s + (0.766 − 0.642i)41-s + (−0.939 − 0.342i)43-s + (0.173 − 0.984i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.173 − 0.984i)17-s + (−0.939 + 0.342i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s + (−0.766 + 0.642i)35-s + 37-s + (0.766 − 0.642i)41-s + (−0.939 − 0.342i)43-s + (0.173 − 0.984i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.939 - 0.341i$
Analytic conductor: \(49.0040\)
Root analytic conductor: \(49.0040\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 456,\ (1:\ ),\ 0.939 - 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.746298693 - 0.3073319832i\)
\(L(\frac12)\) \(\approx\) \(1.746298693 - 0.3073319832i\)
\(L(1)\) \(\approx\) \(1.045496997 - 0.09573713285i\)
\(L(1)\) \(\approx\) \(1.045496997 - 0.09573713285i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−23.91611515787305725492875204210, −22.889126903976249797248314214358, −22.09655011928993903553722136177, −21.35442232717386799521805426351, −20.27525639863875896225545279832, −19.44606598494544484777943884152, −18.63930006831182463159566185279, −17.99304717113100571338574731301, −16.78456274953210678650812300771, −15.86404871956747647352208899102, −15.12062332588199481032698766359, −14.43665640000385806278839607700, −13.23146336416567375715467437068, −12.237794459946709556359520910769, −11.35818677692859113815973394674, −10.84352074629028050544902209456, −9.44743531191786968044093752927, −8.15308819443247027938567278919, −8.10176127513389414688519596660, −6.39636943335503572248799280431, −5.75438093951861296617883959312, −4.30007606550950464999970832232, −3.484160342212334932450980558676, −2.2722661865649626012943152156, −0.7674833544319264692027137319, 0.72820389624505785838449842488, 1.8497610663850626641437536679, 3.60506543486163139266189995822, 4.257579555001768772487405525834, 5.1929045858463001221162728753, 6.82047239027881377321296551296, 7.38261868533948278849690182756, 8.43858848957296486425388269339, 9.34495365463008408265539052747, 10.52944774736787268535824267806, 11.46990872869963470969936038161, 12.072806179057468164624780423404, 13.229209590673629807016333612761, 14.15374566429486767846962696603, 14.96498218719898453419901419260, 16.09469716905914542512070416389, 16.57302172916162554791973863660, 17.73876347541133968043431799737, 18.45301234883060455030591077257, 19.73533877416205782297328466255, 20.1472030350754482550464057413, 20.909754858506592509123145570987, 22.07302359155274964480164209123, 23.14062611999146798962516168708, 23.53512961898691282742445173442

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