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Properties

Label 2-504-63.16-c1-0-18
Degree $2$
Conductor $504$
Sign $0.950 - 0.311i$
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  + (1.34 + 1.09i)3-s + 2.66·5-s + (1.94 − 1.79i)7-s + (0.613 + 2.93i)9-s + 1.36·11-s + (−2.75 − 4.77i)13-s + (3.58 + 2.91i)15-s + (−1.23 − 2.14i)17-s + (−2.19 + 3.80i)19-s + (4.57 − 0.286i)21-s − 4.69·23-s + 2.11·25-s + (−2.38 + 4.61i)27-s + (2.94 − 5.10i)29-s + (−1.55 + 2.69i)31-s + ⋯
L(s)  = 1  + (0.776 + 0.630i)3-s + 1.19·5-s + (0.735 − 0.678i)7-s + (0.204 + 0.978i)9-s + 0.411·11-s + (−0.764 − 1.32i)13-s + (0.925 + 0.752i)15-s + (−0.300 − 0.520i)17-s + (−0.503 + 0.872i)19-s + (0.998 − 0.0626i)21-s − 0.977·23-s + 0.423·25-s + (−0.458 + 0.888i)27-s + (0.547 − 0.948i)29-s + (−0.279 + 0.484i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.950 - 0.311i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.950 - 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24491 + 0.357978i\)
\(L(\frac12)\) \(\approx\) \(2.24491 + 0.357978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 - 1.09i)T \)
7 \( 1 + (-1.94 + 1.79i)T \)
good5 \( 1 - 2.66T + 5T^{2} \)
11 \( 1 - 1.36T + 11T^{2} \)
13 \( 1 + (2.75 + 4.77i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.23 + 2.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.19 - 3.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.69T + 23T^{2} \)
29 \( 1 + (-2.94 + 5.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.55 - 2.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.15 - 5.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.38 - 2.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.87 - 8.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.02 - 8.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.47 + 2.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.77 - 3.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.663 + 1.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.14 - 7.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (1.11 + 1.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.41 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.15 + 8.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.73 + 13.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.55 - 4.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.48442793011327081403196651292, −10.14391113571537606503064241448, −9.378193844826833363830498542199, −8.239074031579735282742921773160, −7.63975082872597650649431826237, −6.24387651932524711418129934027, −5.16563262480047653374311851796, −4.27590374440593461756360694992, −2.90860496116361990194103105867, −1.73310566367429714672396927767, 1.87072224065427162314867571572, 2.27830417728386516664987463868, 4.05345465673540611656840113463, 5.33174851696915635080121793371, 6.41621224607964763127835588882, 7.12356393640237478194012699130, 8.412933034262014333977045802831, 9.053952270951411789681499316563, 9.669276966046941022905315333056, 10.86189169219777165767746513209

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