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Properties

Label 2-504-56.3-c1-0-2
Degree $2$
Conductor $504$
Sign $-0.952 + 0.304i$
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  + (0.321 + 1.37i)2-s + (−1.79 + 0.886i)4-s + (−1.25 − 2.16i)5-s + (1.36 + 2.26i)7-s + (−1.79 − 2.18i)8-s + (2.58 − 2.42i)10-s + (−2.83 + 4.91i)11-s − 5.31·13-s + (−2.68 + 2.60i)14-s + (2.42 − 3.17i)16-s + (0.393 + 0.227i)17-s + (−3.19 + 1.84i)19-s + (4.16 + 2.77i)20-s + (−7.68 − 2.32i)22-s + (−4.43 + 2.56i)23-s + ⋯
L(s)  = 1  + (0.227 + 0.973i)2-s + (−0.896 + 0.443i)4-s + (−0.559 − 0.969i)5-s + (0.515 + 0.857i)7-s + (−0.635 − 0.771i)8-s + (0.816 − 0.765i)10-s + (−0.855 + 1.48i)11-s − 1.47·13-s + (−0.717 + 0.696i)14-s + (0.606 − 0.794i)16-s + (0.0955 + 0.0551i)17-s + (−0.733 + 0.423i)19-s + (0.931 + 0.620i)20-s + (−1.63 − 0.495i)22-s + (−0.924 + 0.533i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.952 + 0.304i$
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: \((2,\ 504,\ (\ :1/2),\ -0.952 + 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0867733 - 0.555676i\)
\(L(\frac12)\) \(\approx\) \(0.0867733 - 0.555676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.321 - 1.37i)T \)
3 \( 1 \)
7 \( 1 + (-1.36 - 2.26i)T \)
good5 \( 1 + (1.25 + 2.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.83 - 4.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.31T + 13T^{2} \)
17 \( 1 + (-0.393 - 0.227i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.19 - 1.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.43 - 2.56i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.57iT - 29T^{2} \)
31 \( 1 + (-3.00 + 5.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.80 - 4.50i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.65iT - 41T^{2} \)
43 \( 1 - 3.66T + 43T^{2} \)
47 \( 1 + (-0.478 - 0.829i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.41 - 3.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.76 - 5.06i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.50 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.65 + 8.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.35iT - 71T^{2} \)
73 \( 1 + (-5.93 - 3.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.71 - 4.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.96iT - 83T^{2} \)
89 \( 1 + (5.91 - 3.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.71iT - 97T^{2} \)
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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

−11.94520876568687106893322420309, −10.19382277852169627716249163868, −9.448246716936821010501546098648, −8.426530199480822010313613256463, −7.85304155329908096796577690705, −7.03561893616814158231169387833, −5.57687206277655612007259476156, −4.91404244373943731434401375901, −4.20687028749760119534121177948, −2.28402367152054128060453803943, 0.29670050623835858437760689206, 2.40094629861024133269446753099, 3.35771339133632018992440667087, 4.41347035565947626658521171621, 5.46159398899787607942811618255, 6.81981887596254955115130237888, 7.85003477295467571801331160848, 8.618259969813667283792928256063, 10.06245063335156122511602147132, 10.55998435721268440676185662930

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