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Properties

Label 2-504-56.3-c1-0-34
Degree $2$
Conductor $504$
Sign $0.0684 + 0.997i$
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  + (1.41 + 0.0681i)2-s + (1.99 + 0.192i)4-s + (−2.08 − 3.61i)5-s + (−2.39 − 1.12i)7-s + (2.79 + 0.407i)8-s + (−2.70 − 5.25i)10-s + (0.855 − 1.48i)11-s − 1.54·13-s + (−3.30 − 1.74i)14-s + (3.92 + 0.766i)16-s + (−2.02 − 1.16i)17-s + (6.09 − 3.52i)19-s + (−3.45 − 7.60i)20-s + (1.30 − 2.03i)22-s + (0.406 − 0.234i)23-s + ⋯
L(s)  = 1  + (0.998 + 0.0482i)2-s + (0.995 + 0.0963i)4-s + (−0.933 − 1.61i)5-s + (−0.905 − 0.423i)7-s + (0.989 + 0.144i)8-s + (−0.854 − 1.66i)10-s + (0.257 − 0.446i)11-s − 0.427·13-s + (−0.884 − 0.466i)14-s + (0.981 + 0.191i)16-s + (−0.490 − 0.282i)17-s + (1.39 − 0.807i)19-s + (−0.773 − 1.69i)20-s + (0.279 − 0.433i)22-s + (0.0846 − 0.0488i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0684 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0684 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0684 + 0.997i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.0684 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47386 - 1.37622i\)
\(L(\frac12)\) \(\approx\) \(1.47386 - 1.37622i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0681i)T \)
3 \( 1 \)
7 \( 1 + (2.39 + 1.12i)T \)
good5 \( 1 + (2.08 + 3.61i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.855 + 1.48i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.54T + 13T^{2} \)
17 \( 1 + (2.02 + 1.16i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.09 + 3.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.406 + 0.234i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.33iT - 29T^{2} \)
31 \( 1 + (-1.58 + 2.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.74 + 4.47i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.31iT - 41T^{2} \)
43 \( 1 + 3.42T + 43T^{2} \)
47 \( 1 + (-2.95 - 5.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.35 - 0.781i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.26 - 3.04i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.55 - 7.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.73 + 6.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.49iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.26i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.46 - 0.843i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.72iT - 83T^{2} \)
89 \( 1 + (1.83 - 1.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.95iT - 97T^{2} \)
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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

−11.14572343634208252532220921716, −9.731578720210418941881268747111, −8.928324277685403016458935017425, −7.77864353069223458757979729366, −7.06430084019912132731270011933, −5.80670018514376064936002364989, −4.82571618683517201052645172953, −4.08918773814015951648718936908, −3.00809096736607721311945041579, −0.902935437191133450352652682727, 2.46291215114026826134587511171, 3.31637250547902977439967717247, 4.12371512019760064398512998241, 5.61572382594772345287262871060, 6.63863994496862265140747503262, 7.11891091166417073030009935138, 8.066713373198754310103498501332, 9.798462681811529201282720417114, 10.33753772902810566936166837841, 11.49248971556877364937360444200

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