Properties

Label 2-504-8.5-c1-0-27
Degree $2$
Conductor $504$
Sign $i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
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)2-s + (0.999 − 1.73i)4-s − 1.41i·5-s + 7-s − 2.82i·8-s + (−1.00 − 1.73i)10-s − 1.41i·11-s + (1.22 − 0.707i)14-s + (−2.00 − 3.46i)16-s − 2.44·17-s + 3.46i·19-s + (−2.44 − 1.41i)20-s + (−1.00 − 1.73i)22-s + 2.44·23-s + 2.99·25-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s − 0.632i·5-s + 0.377·7-s − 0.999i·8-s + (−0.316 − 0.547i)10-s − 0.426i·11-s + (0.327 − 0.188i)14-s + (−0.500 − 0.866i)16-s − 0.594·17-s + 0.794i·19-s + (−0.547 − 0.316i)20-s + (−0.213 − 0.369i)22-s + 0.510·23-s + 0.599·25-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67060 - 1.67060i\)
\(L(\frac12)\) \(\approx\) \(1.67060 - 1.67060i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 2.44T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 3.46iT - 37T^{2} \)
41 \( 1 + 7.34T + 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 7.34T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.88491248071221724939226717262, −10.02156961004943414571443464077, −9.006559617978721619799465921491, −8.050535525008617212301035740878, −6.78347985266239099448637801591, −5.79496995979326825796546263568, −4.87223307419506817932320292005, −3.99966843890866509262109421250, −2.66300379041176097578364095511, −1.20982805244815938810505320924, 2.23893208315553379262902045330, 3.38340487141076566696957292529, 4.59216245939074463375673807545, 5.41732281679631057215162647542, 6.75669555127918664552439288560, 7.09702954877581353712381914170, 8.287355640742835929465707995130, 9.180765534108394816574388074260, 10.64408566649989664700194609127, 11.12566194720480836714014333715

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