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Properties

Label 2-504-504.283-c1-0-9
Degree $2$
Conductor $504$
Sign $-0.955 - 0.294i$
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  + (1.07 + 0.920i)2-s + (0.0180 − 1.73i)3-s + (0.306 + 1.97i)4-s − 4.13·5-s + (1.61 − 1.84i)6-s + (1.26 + 2.32i)7-s + (−1.49 + 2.40i)8-s + (−2.99 − 0.0626i)9-s + (−4.43 − 3.80i)10-s − 4.32·11-s + (3.42 − 0.494i)12-s + (−0.641 + 1.11i)13-s + (−0.776 + 3.66i)14-s + (−0.0747 + 7.16i)15-s + (−3.81 + 1.21i)16-s + (−2.26 − 1.30i)17-s + ⋯
L(s)  = 1  + (0.759 + 0.650i)2-s + (0.0104 − 0.999i)3-s + (0.153 + 0.988i)4-s − 1.84·5-s + (0.658 − 0.752i)6-s + (0.479 + 0.877i)7-s + (−0.526 + 0.849i)8-s + (−0.999 − 0.0208i)9-s + (−1.40 − 1.20i)10-s − 1.30·11-s + (0.989 − 0.142i)12-s + (−0.177 + 0.308i)13-s + (−0.207 + 0.978i)14-s + (−0.0193 + 1.84i)15-s + (−0.953 + 0.302i)16-s + (−0.549 − 0.317i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.955 - 0.294i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.955 - 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0925832 + 0.615656i\)
\(L(\frac12)\) \(\approx\) \(0.0925832 + 0.615656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 - 0.920i)T \)
3 \( 1 + (-0.0180 + 1.73i)T \)
7 \( 1 + (-1.26 - 2.32i)T \)
good5 \( 1 + 4.13T + 5T^{2} \)
11 \( 1 + 4.32T + 11T^{2} \)
13 \( 1 + (0.641 - 1.11i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.26 + 1.30i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.98 + 1.14i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.07iT - 23T^{2} \)
29 \( 1 + (-1.34 + 0.778i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.20 + 5.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.81 - 2.20i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.171 - 0.0990i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.95 + 3.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.57 + 2.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.24 - 2.44i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.47 + 2.58i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.40 - 7.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.06 - 1.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.81iT - 71T^{2} \)
73 \( 1 + (-10.0 - 5.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.50 + 4.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.89 - 2.82i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.48 + 4.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.28 - 3.05i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.65829236360648490292005926492, −11.10746183822919727994674500207, −8.976102254789125356381407149370, −8.184655508728083295505959704137, −7.58383602597187497603576346118, −7.06302721384624603052533498964, −5.64090789856751691895897809844, −4.88599088601755440866945352794, −3.54523027251990746822200714596, −2.47650111326764417902964487867, 0.27515825148396235473430920521, 2.85926635173858507745100970780, 3.79551875106078495963408929160, 4.55253192506237810517313934836, 5.18456486881594744741444768991, 6.84352516695699894240392175419, 7.930153936539904798155792598229, 8.672822252770149856413738622785, 10.18858380390401801311072240617, 10.75653109211542847833985050425

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