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Properties

Label 1-504-504.445-r0-0-0
Degree $1$
Conductor $504$
Sign $0.592 - 0.805i$
Analytic cond. $2.34056$
Root an. cond. $2.34056$
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  − 5-s − 11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 23-s + 25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (0.5 + 0.866i)53-s + 55-s + ⋯
L(s)  = 1  − 5-s − 11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 23-s + 25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (0.5 + 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.592 - 0.805i$
Analytic conductor: \(2.34056\)
Root analytic conductor: \(2.34056\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (0:\ ),\ 0.592 - 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8259947036 - 0.4175216469i\)
\(L(\frac12)\) \(\approx\) \(0.8259947036 - 0.4175216469i\)
\(L(1)\) \(\approx\) \(0.8531112661 - 0.09877766353i\)
\(L(1)\) \(\approx\) \(0.8531112661 - 0.09877766353i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + (0.5 + 0.866i)T \)
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 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
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 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−23.655446137473973451270927784172, −23.03175921240794221890658319720, −22.23669326630852897652671135066, −21.0798932820559949812196007427, −20.36755676654061897452067894329, −19.58679069963466265500385660037, −18.634844824813402910159347767038, −18.01816864161809421163507279833, −16.81727238032137025095457744506, −15.96278485081679750965086096351, −15.27963897069174080533263402238, −14.52682685687533435993831242580, −13.07778466471778557280157038383, −12.73204774396025089363096501919, −11.44576521926785764368801363496, −10.810255923969549777334129830638, −9.85695561711862577031342626475, −8.42118154207769640692268674985, −8.01384694978710106020568697251, −6.9529324834232920650195273897, −5.75175001485831156495494532077, −4.75500982857319301848813797611, −3.63832414625329459249927871459, −2.780007424532864848546682014780, −1.117931979127501033151806049276, 0.60008774172829759382093153915, 2.33937274838053985705165182337, 3.39047804030712567965612645160, 4.49752746220442812117582911140, 5.31328409810932417798297322964, 6.82145470515322671864244899458, 7.40501095982929040431939417382, 8.53657109462618665924509653309, 9.26383344510081283929210865829, 10.63346689943752799419926320631, 11.311713936734720414401078785979, 12.11661531928318290667037121101, 13.192357922449074615533275415895, 13.952464616809367669690354277339, 15.167451322128947571534278171, 15.81355258439741355786653704866, 16.43344923274989608157153847871, 17.68176681596777261284146611379, 18.53234486569291704134397488838, 19.23068837374276714131743310113, 20.14652431248155157734146511202, 20.902562419614151664365862153050, 21.807066729473221842100578558716, 22.89815920474125673719992658813, 23.4574084623884375171844080999

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