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Properties

Label 2-504-63.38-c1-0-22
Degree $2$
Conductor $504$
Sign $-0.562 + 0.826i$
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  + (0.0709 − 1.73i)3-s + (2.04 − 3.54i)5-s + (−2.21 + 1.44i)7-s + (−2.98 − 0.245i)9-s + (3.24 − 1.87i)11-s + (2.68 − 1.54i)13-s + (−5.99 − 3.79i)15-s + (−0.219 + 0.379i)17-s + (−2.68 + 1.54i)19-s + (2.34 + 3.93i)21-s + (−2.43 − 1.40i)23-s + (−5.90 − 10.2i)25-s + (−0.636 + 5.15i)27-s + (0.122 + 0.0704i)29-s + 10.1i·31-s + ⋯
L(s)  = 1  + (0.0409 − 0.999i)3-s + (0.916 − 1.58i)5-s + (−0.836 + 0.547i)7-s + (−0.996 − 0.0818i)9-s + (0.979 − 0.565i)11-s + (0.743 − 0.429i)13-s + (−1.54 − 0.980i)15-s + (−0.0531 + 0.0920i)17-s + (−0.615 + 0.355i)19-s + (0.512 + 0.858i)21-s + (−0.508 − 0.293i)23-s + (−1.18 − 2.04i)25-s + (−0.122 + 0.992i)27-s + (0.0226 + 0.0130i)29-s + 1.81i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.562 + 0.826i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.562 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693177 - 1.31068i\)
\(L(\frac12)\) \(\approx\) \(0.693177 - 1.31068i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0709 + 1.73i)T \)
7 \( 1 + (2.21 - 1.44i)T \)
good5 \( 1 + (-2.04 + 3.54i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.24 + 1.87i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.68 + 1.54i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.219 - 0.379i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.68 - 1.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.43 + 1.40i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.122 - 0.0704i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + (1.72 + 2.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.78 - 3.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.66 + 8.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.53T + 47T^{2} \)
53 \( 1 + (-7.08 - 4.09i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.02T + 59T^{2} \)
61 \( 1 + 15.0iT - 61T^{2} \)
67 \( 1 + 6.30T + 67T^{2} \)
71 \( 1 - 0.881iT - 71T^{2} \)
73 \( 1 + (-10.3 - 5.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.29T + 79T^{2} \)
83 \( 1 + (0.293 - 0.508i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.77 - 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.29 - 4.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.57536121855573644837796678685, −9.383315763137330455743673932759, −8.749478324095255607506073625730, −8.280355552625605118170094666288, −6.63337534787142425360050916815, −6.02676318300877901491238263938, −5.26850152313465965319254239243, −3.65030070081075299941573713870, −2.07653288710482082641543847311, −0.914303258132455530866634033774, 2.29905803712603227387102810650, 3.46622998355586688442020356386, 4.24282791837695875111337198995, 5.97391350241165721403872917307, 6.41416176463821647666027645824, 7.37738084572358219404194678715, 8.950085558911454171425673399576, 9.738348576464805438884074782796, 10.16475888031622106160533856635, 11.04037391012214911447382865742

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