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Properties

Label 2-504-56.3-c1-0-25
Degree $2$
Conductor $504$
Sign $0.999 - 0.0322i$
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  + (0.765 + 1.18i)2-s + (−0.829 + 1.82i)4-s + (−1.61 − 2.79i)5-s + (1.82 − 1.91i)7-s + (−2.79 + 0.406i)8-s + (2.08 − 4.05i)10-s + (1.10 − 1.91i)11-s + 5.08·13-s + (3.67 + 0.709i)14-s + (−2.62 − 3.01i)16-s + (2.73 + 1.57i)17-s + (2.93 − 1.69i)19-s + (6.42 − 0.619i)20-s + (3.12 − 0.150i)22-s + (−2.65 + 1.53i)23-s + ⋯
L(s)  = 1  + (0.541 + 0.840i)2-s + (−0.414 + 0.910i)4-s + (−0.721 − 1.25i)5-s + (0.690 − 0.723i)7-s + (−0.989 + 0.143i)8-s + (0.660 − 1.28i)10-s + (0.333 − 0.577i)11-s + 1.40·13-s + (0.981 + 0.189i)14-s + (−0.656 − 0.754i)16-s + (0.663 + 0.383i)17-s + (0.673 − 0.388i)19-s + (1.43 − 0.138i)20-s + (0.666 − 0.0320i)22-s + (−0.553 + 0.319i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.999 - 0.0322i$
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: \((2,\ 504,\ (\ :1/2),\ 0.999 - 0.0322i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73096 + 0.0279054i\)
\(L(\frac12)\) \(\approx\) \(1.73096 + 0.0279054i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.765 - 1.18i)T \)
3 \( 1 \)
7 \( 1 + (-1.82 + 1.91i)T \)
good5 \( 1 + (1.61 + 2.79i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.10 + 1.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.08T + 13T^{2} \)
17 \( 1 + (-2.73 - 1.57i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.93 + 1.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.65 - 1.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.88iT - 29T^{2} \)
31 \( 1 + (1.01 - 1.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.798 - 0.460i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.96iT - 41T^{2} \)
43 \( 1 + 6.68T + 43T^{2} \)
47 \( 1 + (1.06 + 1.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.12 - 1.80i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.6 - 6.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.34 + 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.40 - 7.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 + (-7.82 - 4.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.60 - 5.54i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.57iT - 83T^{2} \)
89 \( 1 + (-6.32 + 3.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.2iT - 97T^{2} \)
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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

−11.41851450912859212583675265219, −9.854392349047137638834130595319, −8.567744678661485157237304332109, −8.292326450979549774921400851242, −7.43414399289474752947718633791, −6.16619507671861861389663271281, −5.24353782032833317126229241530, −4.22311851224560789859948202201, −3.60870665183944858846350049849, −1.01875878757411270199329750920, 1.68017666539655756104427067222, 3.09013138902403476027723677838, 3.82517233727672832426762007555, 5.12668686418898125249684237220, 6.13888713649850078716254151734, 7.18488965691619257417702862653, 8.323926701698122999261769161423, 9.295217603411637093606315023210, 10.44963310690877407520120574144, 10.97414040303726160486558690191

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