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Properties

Label 2-504-63.38-c1-0-1
Degree $2$
Conductor $504$
Sign $-0.998 - 0.0534i$
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  + (−0.419 + 1.68i)3-s + (1.02 − 1.77i)5-s + (−2.64 − 0.105i)7-s + (−2.64 − 1.41i)9-s + (−5.11 + 2.95i)11-s + (0.139 − 0.0804i)13-s + (2.55 + 2.46i)15-s + (−2.77 + 4.81i)17-s + (−4.02 + 2.32i)19-s + (1.28 − 4.39i)21-s + (0.375 + 0.216i)23-s + (0.400 + 0.694i)25-s + (3.48 − 3.85i)27-s + (−1.95 − 1.12i)29-s − 2.97i·31-s + ⋯
L(s)  = 1  + (−0.242 + 0.970i)3-s + (0.458 − 0.793i)5-s + (−0.999 − 0.0398i)7-s + (−0.882 − 0.470i)9-s + (−1.54 + 0.890i)11-s + (0.0386 − 0.0223i)13-s + (0.658 + 0.636i)15-s + (−0.674 + 1.16i)17-s + (−0.922 + 0.532i)19-s + (0.280 − 0.959i)21-s + (0.0782 + 0.0451i)23-s + (0.0801 + 0.138i)25-s + (0.669 − 0.742i)27-s + (−0.363 − 0.209i)29-s − 0.534i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.998 - 0.0534i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.998 - 0.0534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00930430 + 0.347688i\)
\(L(\frac12)\) \(\approx\) \(0.00930430 + 0.347688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.419 - 1.68i)T \)
7 \( 1 + (2.64 + 0.105i)T \)
good5 \( 1 + (-1.02 + 1.77i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.11 - 2.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.139 + 0.0804i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.77 - 4.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.02 - 2.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.375 - 0.216i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.95 + 1.12i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.97iT - 31T^{2} \)
37 \( 1 + (-2.17 - 3.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.35 + 4.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.82 + 3.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.130T + 47T^{2} \)
53 \( 1 + (10.7 + 6.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.44T + 59T^{2} \)
61 \( 1 - 6.90iT - 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + 1.48iT - 71T^{2} \)
73 \( 1 + (2.60 + 1.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 + (7.62 - 13.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.04 + 7.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.61 + 1.50i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.04992026546777198781451806351, −10.30384681149052229729040817980, −9.739207247126405567108494453198, −8.882004829083388746454694958200, −7.979808711702374331900462918469, −6.51799760755870962729811252303, −5.63283060469113219180900088350, −4.76529534150695070056061234542, −3.74739601057663428986566269250, −2.29773486624907612023118815693, 0.19248082267395231801320911320, 2.46101202824975734801381721872, 3.02555157221128874022351796825, 5.03957269510272630197328383779, 6.11249968483627210164023387586, 6.67940025531089819527510457755, 7.58631728660133603916303300191, 8.604754026088129271036170310027, 9.639249513244584191705016573012, 10.83294737754423946691572821898

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