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Properties

Label 2-504-504.283-c1-0-68
Degree $2$
Conductor $504$
Sign $-0.510 + 0.859i$
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  + (−0.149 − 1.40i)2-s + (0.512 − 1.65i)3-s + (−1.95 + 0.420i)4-s + 3.15·5-s + (−2.40 − 0.473i)6-s + (2.64 + 0.0803i)7-s + (0.884 + 2.68i)8-s + (−2.47 − 1.69i)9-s + (−0.471 − 4.43i)10-s + 1.62·11-s + (−0.306 + 3.45i)12-s + (0.0400 − 0.0693i)13-s + (−0.282 − 3.73i)14-s + (1.61 − 5.21i)15-s + (3.64 − 1.64i)16-s + (−1.13 − 0.658i)17-s + ⋯
L(s)  = 1  + (−0.105 − 0.994i)2-s + (0.296 − 0.955i)3-s + (−0.977 + 0.210i)4-s + 1.41·5-s + (−0.981 − 0.193i)6-s + (0.999 + 0.0303i)7-s + (0.312 + 0.949i)8-s + (−0.824 − 0.565i)9-s + (−0.149 − 1.40i)10-s + 0.491·11-s + (−0.0884 + 0.996i)12-s + (0.0111 − 0.0192i)13-s + (−0.0755 − 0.997i)14-s + (0.417 − 1.34i)15-s + (0.911 − 0.411i)16-s + (−0.276 − 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.510 + 0.859i$
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: \((2,\ 504,\ (\ :1/2),\ -0.510 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.909265 - 1.59735i\)
\(L(\frac12)\) \(\approx\) \(0.909265 - 1.59735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.149 + 1.40i)T \)
3 \( 1 + (-0.512 + 1.65i)T \)
7 \( 1 + (-2.64 - 0.0803i)T \)
good5 \( 1 - 3.15T + 5T^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 + (-0.0400 + 0.0693i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.13 + 0.658i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.49 + 2.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.24iT - 23T^{2} \)
29 \( 1 + (3.48 - 2.01i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.98 + 3.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.45 - 5.45i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.5 + 6.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.79 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.424 - 0.735i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.62 + 2.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.75 - 2.16i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.63 - 4.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.10 + 5.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.99iT - 71T^{2} \)
73 \( 1 + (-7.72 - 4.45i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.5 - 7.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.34 + 1.35i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.04 - 2.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.1 + 8.72i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.77112798739453794361031943314, −9.577134333485937373444035968214, −9.086817970971774721650619755984, −8.121868438652878715933377518222, −7.08243387399823047441064699702, −5.79135102242219061905865136959, −4.98371819894881583854341851058, −3.31705626273818638483684500934, −2.04975275633126386850411330432, −1.39056169216474883263420600842, 1.85958813754712244558470439124, 3.70252664814291539723406420337, 4.92792053558995189520686146547, 5.48080107029064455802830137265, 6.47627969108998940001861754741, 7.71281980025386418730850889601, 8.755925854354669188161795721061, 9.220817368303666502126336305613, 10.17116562113132255795647326675, 10.73998971213969511592224714699

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