Properties

Label 1-72-72.67-r1-0-0
Degree $1$
Conductor $72$
Sign $0.766 + 0.642i$
Analytic cond. $7.73747$
Root an. cond. $7.73747$
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 + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + 17-s + 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + 35-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + 17-s + 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + 35-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(7.73747\)
Root analytic conductor: \(7.73747\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 72,\ (1:\ ),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.745677577 + 0.6353746767i\)
\(L(\frac12)\) \(\approx\) \(1.745677577 + 0.6353746767i\)
\(L(1)\) \(\approx\) \(1.263065016 + 0.2227124407i\)
\(L(1)\) \(\approx\) \(1.263065016 + 0.2227124407i\)

Euler product

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

−31.40916438729701830231600969059, −30.12753641731710659504815192849, −28.92167657720444248972829447617, −28.10594312009296866917038589917, −27.07026158233903293823579314182, −25.547181956857525600145434504796, −24.72848694010798829289649574716, −23.81375358318902749690244378886, −22.30599009759026248389293879837, −21.1537796212054657106138292465, −20.47955372880888938086465315063, −18.82749525458060758769273809375, −17.89752281034223662205894727946, −16.59192368279084311297038268291, −15.5695588138305825180212761129, −14.1150714386743220817540831650, −12.93958269960061628301249253935, −11.838637757136705722088844444069, −10.35110236406855836961010352369, −8.89793845402069964832374016450, −8.019485911085724565308907214931, −5.87127898904817694011200059772, −5.06987474600708054945457132646, −2.96245566030996765290305857002, −1.101560726504607710923192621426, 1.63036367331278909056454026500, 3.42037687110345037306645455509, 5.08648366168555207668888162267, 6.746898685675789959251151415055, 7.741374952112187480015323770866, 9.641231071447980892461308889663, 10.59406520539960375815065004851, 11.830198563980962069457428171093, 13.578738511390696386748458350931, 14.28748392882596993623362620385, 15.621616933279643285793159534903, 17.12171379827086813108326308436, 18.02396237613425544392367353610, 19.14136138093604138617881530797, 20.61817670761550197646771554455, 21.43072641171321631759118402323, 22.89283692363934332560558352731, 23.57206135434059034606913410938, 25.11840314984425902353371873515, 26.138522546024668576217683024360, 26.9218358240940399710870186547, 28.318597120805601945142392832528, 29.436064620534546777552534944209, 30.42526831892212707758848705260, 31.19092655372331936658688498156

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