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Properties

Label 2-504-168.5-c1-0-9
Degree $2$
Conductor $504$
Sign $-0.657 - 0.753i$
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  + (−0.202 + 1.39i)2-s + (−1.91 − 0.567i)4-s + (3.18 + 1.84i)5-s + (−0.998 + 2.45i)7-s + (1.18 − 2.56i)8-s + (−3.22 + 4.08i)10-s + (0.568 + 0.984i)11-s + 3.62·13-s + (−3.22 − 1.89i)14-s + (3.35 + 2.17i)16-s + (−2.84 − 4.93i)17-s + (−2.63 + 4.56i)19-s + (−5.06 − 5.33i)20-s + (−1.49 + 0.596i)22-s + (3.19 + 1.84i)23-s + ⋯
L(s)  = 1  + (−0.143 + 0.989i)2-s + (−0.958 − 0.283i)4-s + (1.42 + 0.822i)5-s + (−0.377 + 0.926i)7-s + (0.418 − 0.908i)8-s + (−1.01 + 1.29i)10-s + (0.171 + 0.296i)11-s + 1.00·13-s + (−0.862 − 0.506i)14-s + (0.838 + 0.544i)16-s + (−0.691 − 1.19i)17-s + (−0.604 + 1.04i)19-s + (−1.13 − 1.19i)20-s + (−0.318 + 0.127i)22-s + (0.665 + 0.384i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.657 - 0.753i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.598661 + 1.31676i\)
\(L(\frac12)\) \(\approx\) \(0.598661 + 1.31676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.202 - 1.39i)T \)
3 \( 1 \)
7 \( 1 + (0.998 - 2.45i)T \)
good5 \( 1 + (-3.18 - 1.84i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.568 - 0.984i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.62T + 13T^{2} \)
17 \( 1 + (2.84 + 4.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.63 - 4.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.19 - 1.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
31 \( 1 + (5.52 - 3.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.63 + 4.98i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 7.00iT - 43T^{2} \)
47 \( 1 + (-3.98 + 6.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.84 - 4.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.813 - 0.469i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.98 - 3.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.18 + 1.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 + (-6.72 + 3.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.68 + 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.89iT - 83T^{2} \)
89 \( 1 + (1.04 - 1.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−10.91329418638530896126610133795, −10.16026876274765536099244872790, −9.174273832722031156969185225721, −8.876224946850853571070021215515, −7.37434746755636768634149793455, −6.47948184322724337001519765706, −5.93696879834276026968991260343, −5.06323498591622782333404533616, −3.39507115650856462761972633560, −1.94045889928102294314200527006, 0.987650910955081586556133593758, 2.14715525751259789307094732332, 3.67292671023602222607885293846, 4.66863696789376551046850247740, 5.79695778882325865103619020769, 6.78990063065033193786981257892, 8.481455583390429600029618560873, 8.911439629056456586506633037929, 9.806801185812511257536995499023, 10.65073925026033589612738590052

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