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Properties

Label 2-504-63.20-c1-0-14
Degree $2$
Conductor $504$
Sign $0.708 + 0.705i$
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  + (0.233 − 1.71i)3-s + (0.0868 + 0.150i)5-s + (1.72 + 2.00i)7-s + (−2.89 − 0.799i)9-s + (3.25 + 1.87i)11-s + (3.54 − 2.04i)13-s + (0.278 − 0.114i)15-s + 6.00·17-s − 6.26i·19-s + (3.84 − 2.48i)21-s + (−6.37 + 3.68i)23-s + (2.48 − 4.30i)25-s + (−2.04 + 4.77i)27-s + (−8.58 − 4.95i)29-s + (−3.76 + 2.17i)31-s + ⋯
L(s)  = 1  + (0.134 − 0.990i)3-s + (0.0388 + 0.0672i)5-s + (0.650 + 0.759i)7-s + (−0.963 − 0.266i)9-s + (0.980 + 0.566i)11-s + (0.982 − 0.567i)13-s + (0.0718 − 0.0294i)15-s + 1.45·17-s − 1.43i·19-s + (0.839 − 0.542i)21-s + (−1.32 + 0.767i)23-s + (0.496 − 0.860i)25-s + (−0.393 + 0.919i)27-s + (−1.59 − 0.920i)29-s + (−0.676 + 0.390i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.708 + 0.705i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.708 + 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55685 - 0.642649i\)
\(L(\frac12)\) \(\approx\) \(1.55685 - 0.642649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.233 + 1.71i)T \)
7 \( 1 + (-1.72 - 2.00i)T \)
good5 \( 1 + (-0.0868 - 0.150i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.25 - 1.87i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.54 + 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.00T + 17T^{2} \)
19 \( 1 + 6.26iT - 19T^{2} \)
23 \( 1 + (6.37 - 3.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.58 + 4.95i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.76 - 2.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.23T + 37T^{2} \)
41 \( 1 + (-0.489 - 0.848i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0468 + 0.0811i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.86 + 3.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.61iT - 53T^{2} \)
59 \( 1 + (0.620 + 1.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.45 - 4.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.21 - 7.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + 15.4iT - 73T^{2} \)
79 \( 1 + (1.21 - 2.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.16 - 5.47i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (5.02 + 2.90i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.16462376411230665871271429915, −9.753656596880729537275981312448, −8.883768696161225150464360903184, −8.047029421324606101932147248522, −7.28507309281912008924211138265, −6.12527916377583089236099191832, −5.47095410608554041700750742554, −3.88436975928200524671292852040, −2.50061072400518810592617575995, −1.28097562439884982842382935483, 1.50482379645221125878025699763, 3.66307118252224580037404854165, 3.93728495221447154215986254869, 5.38150923818403508835727130223, 6.18949571466381176750082530871, 7.64829932784069265153807646596, 8.416457865884510590248641450156, 9.342925390771496764056450826922, 10.13514223040713759865855759434, 11.05349008204579704509370738442

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