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Properties

Label 2-504-168.107-c1-0-27
Degree $2$
Conductor $504$
Sign $-0.00942 + 0.999i$
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  + (1.23 − 0.688i)2-s + (1.05 − 1.70i)4-s + (−0.317 + 0.550i)5-s + (−2.11 − 1.58i)7-s + (0.130 − 2.82i)8-s + (−0.0138 + 0.898i)10-s + (3.16 − 1.82i)11-s − 4.15i·13-s + (−3.70 − 0.499i)14-s + (−1.78 − 3.58i)16-s + (3.01 − 1.74i)17-s + (−1.99 + 3.45i)19-s + (0.601 + 1.11i)20-s + (2.65 − 4.43i)22-s + (1.47 − 2.54i)23-s + ⋯
L(s)  = 1  + (0.873 − 0.486i)2-s + (0.526 − 0.850i)4-s + (−0.142 + 0.246i)5-s + (−0.800 − 0.598i)7-s + (0.0462 − 0.998i)8-s + (−0.00438 + 0.284i)10-s + (0.954 − 0.550i)11-s − 1.15i·13-s + (−0.991 − 0.133i)14-s + (−0.445 − 0.895i)16-s + (0.732 − 0.422i)17-s + (−0.457 + 0.791i)19-s + (0.134 + 0.250i)20-s + (0.565 − 0.945i)22-s + (0.306 − 0.530i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00942 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00942 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.00942 + 0.999i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.00942 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52730 - 1.54177i\)
\(L(\frac12)\) \(\approx\) \(1.52730 - 1.54177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 + 0.688i)T \)
3 \( 1 \)
7 \( 1 + (2.11 + 1.58i)T \)
good5 \( 1 + (0.317 - 0.550i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.16 + 1.82i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.15iT - 13T^{2} \)
17 \( 1 + (-3.01 + 1.74i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.99 - 3.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.47 + 2.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.35T + 29T^{2} \)
31 \( 1 + (5.20 - 3.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.59 + 0.923i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.4iT - 41T^{2} \)
43 \( 1 + 2.83T + 43T^{2} \)
47 \( 1 + (4.61 - 7.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.99 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.10 + 5.25i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.72 + 0.995i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.01 + 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.737T + 71T^{2} \)
73 \( 1 + (-2.13 - 3.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.74 - 4.47i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.67iT - 83T^{2} \)
89 \( 1 + (4.63 + 2.67i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−10.71034467962250406410723477634, −10.14225706514427145514691201814, −9.178384129420574033655580915194, −7.81161155880353643536323586352, −6.72186245995346518098569662744, −6.05127360459788717731080229306, −4.88130486476249406465925083199, −3.58895594240553910545562473551, −3.05577016037458247158950444901, −1.07153052529178441681084321939, 2.14692201241232259717707443735, 3.54283409145519058148519923209, 4.44147798510137838249573687174, 5.55192289512652624722523763554, 6.61456073894643759912155625294, 7.07438946154711689205631254696, 8.532902126926063301327597049045, 9.113247556295643963026725725136, 10.26555559419426612530628641782, 11.63617706714145850341102350264

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