Properties

Label 2-504-63.38-c1-0-11
Degree $2$
Conductor $504$
Sign $0.955 - 0.293i$
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  + (−1.14 + 1.29i)3-s + (−0.527 + 0.914i)5-s + (0.781 − 2.52i)7-s + (−0.365 − 2.97i)9-s + (5.40 − 3.12i)11-s + (0.872 − 0.503i)13-s + (−0.580 − 1.73i)15-s + (−3.26 + 5.66i)17-s + (1.73 − 1.00i)19-s + (2.38 + 3.91i)21-s + (3.81 + 2.20i)23-s + (1.94 + 3.36i)25-s + (4.28 + 2.94i)27-s + (6.12 + 3.53i)29-s − 2.39i·31-s + ⋯
L(s)  = 1  + (−0.662 + 0.748i)3-s + (−0.236 + 0.408i)5-s + (0.295 − 0.955i)7-s + (−0.121 − 0.992i)9-s + (1.63 − 0.941i)11-s + (0.241 − 0.139i)13-s + (−0.149 − 0.447i)15-s + (−0.792 + 1.37i)17-s + (0.397 − 0.229i)19-s + (0.519 + 0.854i)21-s + (0.794 + 0.458i)23-s + (0.388 + 0.672i)25-s + (0.823 + 0.566i)27-s + (1.13 + 0.657i)29-s − 0.429i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.955 - 0.293i$
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: \((2,\ 504,\ (\ :1/2),\ 0.955 - 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21613 + 0.182605i\)
\(L(\frac12)\) \(\approx\) \(1.21613 + 0.182605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 - 1.29i)T \)
7 \( 1 + (-0.781 + 2.52i)T \)
good5 \( 1 + (0.527 - 0.914i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.40 + 3.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.872 + 0.503i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.26 - 5.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 + 1.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.81 - 2.20i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.12 - 3.53i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 + (3.64 + 6.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.80 + 3.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.60 + 2.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.74T + 47T^{2} \)
53 \( 1 + (-6.02 - 3.47i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 + 8.20iT - 61T^{2} \)
67 \( 1 - 0.122T + 67T^{2} \)
71 \( 1 + 5.37iT - 71T^{2} \)
73 \( 1 + (-14.4 - 8.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 8.86T + 79T^{2} \)
83 \( 1 + (-1.07 + 1.86i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.23 - 3.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.960 + 0.554i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.04881908912786791215955189718, −10.34479856436575718117480226202, −9.196572803580900969592697118739, −8.497204228729789829074248037199, −7.00793168351614791954243880330, −6.44827282524923414299513009296, −5.28124453378794356379983880941, −3.99474215624539399088707943776, −3.54224432955483715429151197962, −1.08617715834011482682476934549, 1.20001240248363065200413033329, 2.54389653679506769796634175559, 4.46544949551468499871861761979, 5.15322861851303430841870895098, 6.51546806613770012724196256827, 6.92113523280126964785163513325, 8.269015338913455766600394023654, 8.961274533267555123584616489147, 9.936066474404010365064282860795, 11.29011123696268009862038959975

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