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Properties

Label 2-504-168.83-c0-0-1
Degree $2$
Conductor $504$
Sign $0.169 - 0.985i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + i·7-s + (−0.707 + 0.707i)8-s − 1.41i·11-s + (−0.707 + 0.707i)14-s − 1.00·16-s + (1.00 − 1.00i)22-s − 1.41·23-s + 25-s − 1.00·28-s + 1.41·29-s + (−0.707 − 0.707i)32-s − 2i·37-s + 1.41·44-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + i·7-s + (−0.707 + 0.707i)8-s − 1.41i·11-s + (−0.707 + 0.707i)14-s − 1.00·16-s + (1.00 − 1.00i)22-s − 1.41·23-s + 25-s − 1.00·28-s + 1.41·29-s + (−0.707 − 0.707i)32-s − 2i·37-s + 1.41·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.169 - 0.985i$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ 0.169 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.202835113\)
\(L(\frac12)\) \(\approx\) \(1.202835113\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.57944660976374549348737807346, −10.63814917999106559165594636977, −9.187575402947078960247140622507, −8.528876386654543243553871521792, −7.76229341550024307099609958232, −6.41640413296609098781640626571, −5.86914721108868346164710145062, −4.89544325237358754597032774619, −3.60173891267830790149977700977, −2.55031771968709636529691255159, 1.56974987402061318053921220907, 3.01076748836098431101843715498, 4.30620581347416844346422667419, 4.83375575350071086980997785005, 6.31370563606752032929309741296, 7.06485411918232500301385965945, 8.241707418754466703895218083264, 9.671009090262501788762736373191, 10.12834567598797130212188382852, 10.90944344399717617207517728970

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