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Properties

Label 2-504-21.2-c0-0-0
Degree $2$
Conductor $504$
Sign $0.987 - 0.154i$
Analytic cond. $0.251528$
Root an. cond. $0.501526$
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  + (1.22 + 0.707i)5-s + (0.5 − 0.866i)7-s + (−1.22 + 0.707i)11-s − 13-s + (0.5 − 0.866i)19-s + (0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (1.22 − 0.707i)35-s + (−0.5 + 0.866i)37-s − 1.41i·41-s − 43-s + (−1.22 − 0.707i)47-s + (−0.499 − 0.866i)49-s − 2·55-s + (−1.22 − 0.707i)65-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)5-s + (0.5 − 0.866i)7-s + (−1.22 + 0.707i)11-s − 13-s + (0.5 − 0.866i)19-s + (0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (1.22 − 0.707i)35-s + (−0.5 + 0.866i)37-s − 1.41i·41-s − 43-s + (−1.22 − 0.707i)47-s + (−0.499 − 0.866i)49-s − 2·55-s + (−1.22 − 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.987 - 0.154i$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ 0.987 - 0.154i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.016419313\)
\(L(\frac12)\) \(\approx\) \(1.016419313\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.89693422058892803480209888636, −10.11171652615922084188038313910, −9.847501068545743014491413352719, −8.432619733096172378865919988009, −7.27628823452008392492026722041, −6.82908323287905615539422532034, −5.38915805438657560386696987233, −4.73803521666916912168467149620, −3.00187391093175300179302104255, −1.96149439826049507956339069820, 1.82436622802639729639815280514, 2.88972272667109435363341588269, 4.84167906884327546955225733171, 5.43949625678711008288418486788, 6.18462475321189764777263888274, 7.76285478212202340210392691558, 8.421934090284460485568408243910, 9.482865181644402017405005818633, 9.993111719736568541048186599210, 11.10031976697460621837210600289

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