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Properties

Label 2-504-56.53-c1-0-6
Degree $2$
Conductor $504$
Sign $-0.561 - 0.827i$
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  + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + (−1.37 + 0.792i)5-s + (1.62 + 2.09i)7-s + 2.82i·8-s + (1.12 − 1.94i)10-s + (−0.148 − 0.0857i)11-s + (−3.46 − 1.41i)14-s + (−2.00 − 3.46i)16-s + 3.17i·20-s + 0.242·22-s + (−1.24 + 2.15i)25-s + (5.24 − 0.717i)28-s + 10.4i·29-s + (−4.62 + 8.00i)31-s + (4.89 + 2.82i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (0.499 − 0.866i)4-s + (−0.614 + 0.354i)5-s + (0.612 + 0.790i)7-s + 0.999i·8-s + (0.354 − 0.614i)10-s + (−0.0448 − 0.0258i)11-s + (−0.925 − 0.377i)14-s + (−0.500 − 0.866i)16-s + 0.709i·20-s + 0.0517·22-s + (−0.248 + 0.430i)25-s + (0.990 − 0.135i)28-s + 1.93i·29-s + (−0.830 + 1.43i)31-s + (0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.561 - 0.827i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.561 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.333843 + 0.630005i\)
\(L(\frac12)\) \(\approx\) \(0.333843 + 0.630005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-1.62 - 2.09i)T \)
good5 \( 1 + (1.37 - 0.792i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.148 + 0.0857i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 10.4iT - 29T^{2} \)
31 \( 1 + (4.62 - 8.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.72 - 5.03i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.6 + 7.32i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.86 - 6.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.8iT - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−11.05918226982730591743415316574, −10.39711138199263322578301735135, −9.136932026461322320457260884631, −8.623372779291531729093484029454, −7.62782165156730878497079257920, −6.94203423214803657211730330463, −5.74747403495618972054661438883, −4.87779702045372470564983419376, −3.16058536683034585584661084168, −1.66470680321015291575887856930, 0.57500521998528341156649605637, 2.14551048995455788175508291742, 3.74558361857957389626030641954, 4.50996030169763667359024946230, 6.16395113507878136653443763787, 7.50535636200627322739403691755, 7.85320456886244068229032305294, 8.818332327631961077126693958940, 9.808730924570214904044865206077, 10.58695968778498176558597543650

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