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Properties

Label 2-504-63.58-c1-0-7
Degree $2$
Conductor $504$
Sign $0.337 - 0.941i$
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  + (−1.47 + 0.900i)3-s + (1.26 + 2.18i)5-s + (2.63 + 0.275i)7-s + (1.37 − 2.66i)9-s + (2.85 − 4.94i)11-s + (−2.45 + 4.24i)13-s + (−3.83 − 2.09i)15-s + (2.49 + 4.32i)17-s + (−0.00383 + 0.00664i)19-s + (−4.14 + 1.96i)21-s + (−0.333 − 0.578i)23-s + (−0.682 + 1.18i)25-s + (0.355 + 5.18i)27-s + (3.85 + 6.66i)29-s − 7.76·31-s + ⋯
L(s)  = 1  + (−0.854 + 0.519i)3-s + (0.564 + 0.977i)5-s + (0.994 + 0.104i)7-s + (0.459 − 0.887i)9-s + (0.861 − 1.49i)11-s + (−0.680 + 1.17i)13-s + (−0.989 − 0.541i)15-s + (0.605 + 1.04i)17-s + (−0.000880 + 0.00152i)19-s + (−0.903 + 0.427i)21-s + (−0.0696 − 0.120i)23-s + (−0.136 + 0.236i)25-s + (0.0685 + 0.997i)27-s + (0.715 + 1.23i)29-s − 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.337 - 0.941i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.337 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09723 + 0.772179i\)
\(L(\frac12)\) \(\approx\) \(1.09723 + 0.772179i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.47 - 0.900i)T \)
7 \( 1 + (-2.63 - 0.275i)T \)
good5 \( 1 + (-1.26 - 2.18i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.85 + 4.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.45 - 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.49 - 4.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.00383 - 0.00664i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.333 + 0.578i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.85 - 6.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.76T + 31T^{2} \)
37 \( 1 + (3.19 - 5.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.21 + 9.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.42 - 7.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.16T + 47T^{2} \)
53 \( 1 + (3.69 + 6.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.523T + 59T^{2} \)
61 \( 1 + 8.99T + 61T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 + 5.68T + 71T^{2} \)
73 \( 1 + (1.52 + 2.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.16T + 79T^{2} \)
83 \( 1 + (0.258 + 0.448i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.19 + 2.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.32 - 7.49i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.92010667660709612051665896516, −10.57046521296916126825446878903, −9.396789607992462532860035183718, −8.611881500943537026887625870703, −7.20299702934705461238133264689, −6.32127667542753242827688847243, −5.63750103457749486362104038951, −4.44925350673478021713325096064, −3.33113011245253427310566910531, −1.58693919327815009778939459538, 1.03351605800107370330502166820, 2.15056853388949290464857065433, 4.46185186332358397780729651366, 5.08901687356473105370283487013, 5.87480894562451668196525939233, 7.32533696379770477946531925546, 7.70297946916329097016221146076, 9.100801366510990147186369285497, 9.873758022874513518386251846995, 10.78588095462965303522102980655

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