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Properties

Label 2-504-504.461-c1-0-47
Degree $2$
Conductor $504$
Sign $-0.311 - 0.950i$
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 + (0.420 + 1.68i)3-s + (0.999 + 1.73i)4-s + (1.99 − 1.15i)5-s + (−0.672 + 2.35i)6-s + (−1.32 + 2.29i)7-s + 2.82i·8-s + (−2.64 + 1.41i)9-s + 3.25·10-s + (−2.48 + 2.40i)12-s + (−2.79 − 4.84i)13-s + (−3.24 + 1.87i)14-s + (2.77 + 2.86i)15-s + (−2.00 + 3.46i)16-s + (−4.24 − 0.138i)18-s + 5.17·19-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.242 + 0.970i)3-s + (0.499 + 0.866i)4-s + (0.891 − 0.514i)5-s + (−0.274 + 0.961i)6-s + (−0.499 + 0.866i)7-s + 0.999i·8-s + (−0.881 + 0.471i)9-s + 1.02·10-s + (−0.718 + 0.695i)12-s + (−0.775 − 1.34i)13-s + (−0.866 + 0.499i)14-s + (0.715 + 0.739i)15-s + (−0.500 + 0.866i)16-s + (−0.999 − 0.0327i)18-s + 1.18·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.311 - 0.950i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54917 + 2.13717i\)
\(L(\frac12)\) \(\approx\) \(1.54917 + 2.13717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.420 - 1.68i)T \)
7 \( 1 + (1.32 - 2.29i)T \)
good5 \( 1 + (-1.99 + 1.15i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.79 + 4.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.17T + 19T^{2} \)
23 \( 1 + (-7.74 + 4.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (12.4 - 7.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.89 + 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (-8.67 + 15.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.21 - 0.700i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 + 84.0i)T^{2} \)
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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

−11.23441898599900227988888881766, −10.18207394648141146117525199388, −9.340016605782548840485392841271, −8.634257383558782798796170087502, −7.54569105784053070682013428503, −6.17618716139600767875909387856, −5.29807521527927040827433483833, −4.94750852891521487679039736395, −3.29547181635058276858407860345, −2.57135031665839240955436088606, 1.34099273667974788335563276012, 2.53638610167321929129841725018, 3.51290767727784088066598726889, 4.98147414376609037385613761574, 6.11080419025144950467599588758, 6.91410329246668506087103645066, 7.39278967684121192124246451637, 9.285894029002264129099195303134, 9.749129094892282532343817648552, 10.88359323118363040811241599632

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