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Properties

Label 2-504-63.5-c1-0-4
Degree $2$
Conductor $504$
Sign $-0.419 - 0.907i$
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  + (0.690 + 1.58i)3-s + (1.10 + 1.91i)5-s + (0.234 + 2.63i)7-s + (−2.04 + 2.19i)9-s + (0.157 + 0.0909i)11-s + (−2.50 − 1.44i)13-s + (−2.27 + 3.07i)15-s + (−1.98 − 3.43i)17-s + (0.867 + 0.500i)19-s + (−4.02 + 2.19i)21-s + (4.86 − 2.80i)23-s + (0.0605 − 0.104i)25-s + (−4.89 − 1.73i)27-s + (0.703 − 0.406i)29-s + 7.96i·31-s + ⋯
L(s)  = 1  + (0.398 + 0.917i)3-s + (0.493 + 0.855i)5-s + (0.0885 + 0.996i)7-s + (−0.682 + 0.731i)9-s + (0.0475 + 0.0274i)11-s + (−0.694 − 0.400i)13-s + (−0.587 + 0.793i)15-s + (−0.481 − 0.833i)17-s + (0.198 + 0.114i)19-s + (−0.878 + 0.478i)21-s + (1.01 − 0.585i)23-s + (0.0121 − 0.0209i)25-s + (−0.942 − 0.334i)27-s + (0.130 − 0.0754i)29-s + 1.43i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.419 - 0.907i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.419 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874232 + 1.36690i\)
\(L(\frac12)\) \(\approx\) \(0.874232 + 1.36690i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.690 - 1.58i)T \)
7 \( 1 + (-0.234 - 2.63i)T \)
good5 \( 1 + (-1.10 - 1.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.157 - 0.0909i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.50 + 1.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.98 + 3.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.867 - 0.500i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.86 + 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.703 + 0.406i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.96iT - 31T^{2} \)
37 \( 1 + (-1.25 + 2.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.612 + 1.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.47 - 9.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.15T + 47T^{2} \)
53 \( 1 + (-1.75 + 1.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.55T + 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 - 6.89T + 67T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (10.1 - 5.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (-7.19 - 12.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.11 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.01 + 1.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.94600881953066158767646993054, −10.33559847497111015430773765877, −9.377138262486188276781749775620, −8.838126216931429930630905151374, −7.66318070616358753102258584959, −6.54555621717675846884865961625, −5.45221188066457406472783814473, −4.63542299078419337352462939594, −3.01832664645336189473233375098, −2.47823561961587200408961098438, 0.963718531219959311132149874758, 2.18372313336380918215096517800, 3.76212974701197599077163092029, 4.95234116201448246823153985529, 6.11613808808540871217171207974, 7.12725075649035508054791570676, 7.81649144031480346027659292206, 8.909118690160930543945997725969, 9.485492958388062049441734389902, 10.66974224911277700079438222806

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