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Properties

Label 1-504-504.499-r1-0-0
Degree $1$
Conductor $504$
Sign $-0.971 - 0.235i$
Analytic cond. $54.1623$
Root an. cond. $54.1623$
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)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s − 31-s + (0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s − 47-s + (0.5 − 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s − 31-s + (0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s − 47-s + (0.5 − 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.971 - 0.235i$
Analytic conductor: \(54.1623\)
Root analytic conductor: \(54.1623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (1:\ ),\ -0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1060975887 - 0.8877097915i\)
\(L(\frac12)\) \(\approx\) \(0.1060975887 - 0.8877097915i\)
\(L(1)\) \(\approx\) \(0.9302193625 - 0.2732293201i\)
\(L(1)\) \(\approx\) \(0.9302193625 - 0.2732293201i\)

Euler product

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

−23.479145134737594855374168158923, −23.04165768041856410407173088653, −22.17550823245304916288479761227, −21.32213912277338833123096985033, −20.47250895451851552537781989216, −19.657256412879055999236844070625, −18.338850727050448709448200441788, −18.15401111202464662859235931830, −17.17342245482269591347545201822, −16.01495079503467334164990407848, −15.15371870811089226038426511671, −14.47392551785172034482027922709, −13.41275206290618592699332011365, −12.7467016839597521294135872228, −11.51359274374494616232614765487, −10.612316444176924668956023436592, −9.96938718235684497501796845716, −8.93397012238669795619001743377, −7.67875775483056061793493296849, −6.964321401879708826031171009097, −5.88210139991836137754209502568, −4.99721183068916288697543669156, −3.5977880922536754697040272099, −2.6514671322110635465729016778, −1.564405580668231109490084182792, 0.21479026322947854863344227523, 1.479496276722120365617550748563, 2.59425373797827232132635037655, 4.02510253923039506493291139336, 4.91023426878184583138563760222, 5.98110597894764359034082632582, 6.76300197928112401531451660786, 8.37120902254327457002888242111, 8.685730570298402286575322421991, 9.798394384874493533502688287950, 10.8495705525102685964965374932, 11.65879897329692419891242270810, 12.96221094289908914267711391393, 13.2893099018057748846648122517, 14.3264999496105613328141223571, 15.42973370844954758974578334015, 16.35740247537089557344744081211, 16.94157623501773044139692094079, 17.88059872552565721727744447559, 18.841365481942039576503925214521, 19.653221949845835104782708017746, 20.64011096768055274443111698066, 21.39904971867326736013949871453, 21.8854397934473475467459837448, 23.224043238033096441010157701599

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