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Properties

Label 1-72-72.29-r1-0-0
Degree $1$
Conductor $72$
Sign $-0.642 - 0.766i$
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

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

Invariants

Degree: \(1\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(7.73747\)
Root analytic conductor: \(7.73747\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 72,\ (1:\ ),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2464516952 - 0.5285173660i\)
\(L(\frac12)\) \(\approx\) \(0.2464516952 - 0.5285173660i\)
\(L(1)\) \(\approx\) \(0.7292309272 - 0.1285830876i\)
\(L(1)\) \(\approx\) \(0.7292309272 - 0.1285830876i\)

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

−31.33816765057265948657028144066, −31.261379570712443886354722830929, −29.36158464739641785688926857959, −28.36227190348329599162538758019, −27.75110406351776935699382474693, −26.18311224488381485006889801226, −25.24367831979855411211771531534, −24.05080278794069172920281652906, −23.13319123428032980961110407147, −21.77551930685614350873765997917, −20.68913660127655506202593361556, −19.56557329187023301823663626792, −18.50314773245295014463399304073, −17.10178493776695212447335825768, −15.888019408777505971879821825566, −15.116958560494163518727844529694, −13.248916306512317853321177030475, −12.41787325712974320205590692161, −11.193696365929633450979002541091, −9.41394751370787855309455340901, −8.55848332628303009177429000596, −6.952061741311805215844667049271, −5.359351407905197843229220904511, −4.01622104871305997660880673099, −2.03050722491665800089732416894, 0.27916258375206402860741786269, 2.85360662224328524951619800970, 4.09044499945765741014199177242, 6.10716495282425505885013601827, 7.272743079556181740697003556942, 8.58456323433546704586149691284, 10.49560746384215428801756113553, 10.982045826259326076251787980679, 12.80483613200410822700470834326, 13.86301526922421918116580714653, 15.21406030141763755971769573511, 16.21491733750108355511825043979, 17.599868931216853304629813732149, 18.83605536408663314060370627049, 19.72543107989759407762863886028, 20.9956782784547701349659788782, 22.433788639640670503583600643966, 23.1356160962578843158485408964, 24.27252356467162525837197831430, 25.82188306221973995012455746989, 26.57978779530438895528603755859, 27.4980391174521508955933955600, 29.01728381798350693365328633522, 29.9253246644374674658240703520, 30.83037668619064299373213638249

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