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Properties

Label 2-504-63.20-c1-0-10
Degree $2$
Conductor $504$
Sign $0.808 - 0.588i$
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  + (1.27 + 1.17i)3-s + (0.00869 + 0.0150i)5-s + (−0.514 − 2.59i)7-s + (0.253 + 2.98i)9-s + (4.13 + 2.38i)11-s + (1.31 − 0.759i)13-s + (−0.00655 + 0.0294i)15-s + 1.91·17-s + 6.45i·19-s + (2.38 − 3.91i)21-s + (3.82 − 2.20i)23-s + (2.49 − 4.32i)25-s + (−3.17 + 4.11i)27-s + (1.73 + 1.00i)29-s + (−6.51 + 3.76i)31-s + ⋯
L(s)  = 1  + (0.736 + 0.676i)3-s + (0.00389 + 0.00673i)5-s + (−0.194 − 0.980i)7-s + (0.0846 + 0.996i)9-s + (1.24 + 0.719i)11-s + (0.364 − 0.210i)13-s + (−0.00169 + 0.00759i)15-s + 0.463·17-s + 1.48i·19-s + (0.520 − 0.853i)21-s + (0.797 − 0.460i)23-s + (0.499 − 0.865i)25-s + (−0.611 + 0.791i)27-s + (0.322 + 0.186i)29-s + (−1.17 + 0.676i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.808 - 0.588i$
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: \((2,\ 504,\ (\ :1/2),\ 0.808 - 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81417 + 0.590811i\)
\(L(\frac12)\) \(\approx\) \(1.81417 + 0.590811i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 - 1.17i)T \)
7 \( 1 + (0.514 + 2.59i)T \)
good5 \( 1 + (-0.00869 - 0.0150i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.13 - 2.38i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.31 + 0.759i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.91T + 17T^{2} \)
19 \( 1 - 6.45iT - 19T^{2} \)
23 \( 1 + (-3.82 + 2.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.73 - 1.00i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.51 - 3.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.10T + 37T^{2} \)
41 \( 1 + (4.67 + 8.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.46 - 2.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.45 + 2.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.94iT - 53T^{2} \)
59 \( 1 + (-4.08 - 7.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.484 - 0.279i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.69 + 6.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + 12.8iT - 73T^{2} \)
79 \( 1 + (-4.75 + 8.24i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.67 + 11.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.00T + 89T^{2} \)
97 \( 1 + (4.09 + 2.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.51935999040337213055014753373, −10.30389281936697956429199116456, −9.234067578385401793668285742960, −8.473159621724442051940010862620, −7.42332554504233116175593655081, −6.59167061063067140629161227685, −5.12760746575604244310612222460, −4.01862090597614413629980101808, −3.39919964503730375378762861950, −1.62490009060128024557554668705, 1.33765997598285492394196943465, 2.79263982810665609441998426189, 3.70042902969869942950648786627, 5.31915868625087476832251510228, 6.41602482737329907168913440761, 7.09105385060564294788531337256, 8.330582035128771076064518578984, 9.088089169740953253780291783045, 9.427716581244125244214968995851, 11.12750007291917852908614674056

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