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Properties

Label 1-72-72.13-r0-0-0
Degree $1$
Conductor $72$
Sign $0.766 + 0.642i$
Analytic cond. $0.334366$
Root an. cond. $0.334366$
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

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

Invariants

Degree: \(1\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.334366\)
Root analytic conductor: \(0.334366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 72,\ (0:\ ),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9197214877 + 0.3347512453i\)
\(L(\frac12)\) \(\approx\) \(0.9197214877 + 0.3347512453i\)
\(L(1)\) \(\approx\) \(1.035205736 + 0.2134755561i\)
\(L(1)\) \(\approx\) \(1.035205736 + 0.2134755561i\)

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

−31.71905557787174441538596254243, −30.17375131833142524894927598414, −29.48421580941850672573034809243, −28.163561540372382389635507260242, −27.42542855836235172091111798517, −25.72186739323419272356986733161, −25.26132519103447050704413751036, −23.72435340757309339707721143610, −22.9122346915152802393354639874, −21.46661557659550218001629314767, −20.36519722341295997147191642723, −19.60071893007075376862556111168, −17.85631056047438788033707824402, −16.983241051665548410315856266625, −15.937349668853705008737854827317, −14.39184439348052417964155640728, −13.1595509650851485054737479288, −12.3133561253029776154034714594, −10.484692531373141296415547819509, −9.52242381613333263134807422902, −8.08358848300956800406786806195, −6.57991586841707826170401745110, −5.10629851236419492999352337039, −3.629394625628101399085476972088, −1.42467481873239634771439473912, 2.22597710360548271714684502500, 3.669186280593918975584787366677, 5.82255136630110305200079307601, 6.6234689328207247566273298344, 8.491082575023549961928390227699, 9.69775569278157298222034893172, 11.007121727582652084083529766350, 12.23350348113595704231288338606, 13.73846673582274412699715473714, 14.67400468122114456720768227004, 16.00173978074129690032718703065, 17.2041748756147054666435420123, 18.81551373666136698933800628125, 18.98622330756815119322015081036, 21.01379150198451652054836439490, 21.85934278670287855152163372819, 22.780363239488029427159042612252, 24.17102377076190764350303422985, 25.44082151383310473372851304468, 26.10979805331162340313045100797, 27.41517636242089709720143388095, 28.6139255685274555331863775509, 29.619248052843133548409247498817, 30.557319028048212582972783100226, 31.81345037823403048832936559512

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