Properties

Label 2-504-9.4-c1-0-7
Degree $2$
Conductor $504$
Sign $0.598 - 0.801i$
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  + (1.58 + 0.687i)3-s + (−0.300 + 0.520i)5-s + (0.5 + 0.866i)7-s + (2.05 + 2.18i)9-s + (0.800 + 1.38i)11-s + (−0.165 + 0.286i)13-s + (−0.834 + 0.620i)15-s − 1.44·17-s + 2.57·19-s + (0.199 + 1.72i)21-s + (−0.924 + 1.60i)23-s + (2.31 + 4.01i)25-s + (1.76 + 4.88i)27-s + (−1.75 − 3.04i)29-s + (4.81 − 8.33i)31-s + ⋯
L(s)  = 1  + (0.917 + 0.396i)3-s + (−0.134 + 0.232i)5-s + (0.188 + 0.327i)7-s + (0.685 + 0.728i)9-s + (0.241 + 0.417i)11-s + (−0.0458 + 0.0793i)13-s + (−0.215 + 0.160i)15-s − 0.351·17-s + 0.591·19-s + (0.0435 + 0.375i)21-s + (−0.192 + 0.333i)23-s + (0.463 + 0.803i)25-s + (0.339 + 0.940i)27-s + (−0.326 − 0.565i)29-s + (0.863 − 1.49i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.598 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73486 + 0.869581i\)
\(L(\frac12)\) \(\approx\) \(1.73486 + 0.869581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.58 - 0.687i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.300 - 0.520i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.800 - 1.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.165 - 0.286i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.44T + 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 + (0.924 - 1.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.75 + 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.81 + 8.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.600T + 37T^{2} \)
41 \( 1 + (3.31 - 5.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.81 + 3.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.95 + 3.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 + (-6.93 + 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.59 - 4.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.90 + 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.17T + 71T^{2} \)
73 \( 1 - 4.13T + 73T^{2} \)
79 \( 1 + (4.06 + 7.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.78 - 4.83i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.83T + 89T^{2} \)
97 \( 1 + (-0.974 - 1.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.06065457777215122309986117411, −9.859003540712379400774942057749, −9.427276179538353006916112949517, −8.343299546177209055565712416268, −7.63144900985958841492917654532, −6.60932612144543570425636573096, −5.22770603655889598112775101207, −4.20408709121265937108293377693, −3.12389423794832267151848038583, −1.91813160399567189891598465733, 1.21493860260094065086161520857, 2.74277733912044740386131447705, 3.83656885352081999245845223984, 4.95495717128829265056161951895, 6.42175640834086552801502056504, 7.21800919589424456466248493210, 8.255224135087342022993990377975, 8.778640244435453988143116202373, 9.806081909362102679777744490480, 10.69597673350873703206910201563

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