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Properties

Label 2-504-63.38-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.838 - 0.544i$
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  + (−1.69 − 0.343i)3-s + (−0.0262 + 0.0455i)5-s + (−2.10 − 1.60i)7-s + (2.76 + 1.16i)9-s + (2.15 − 1.24i)11-s + (−2.51 + 1.45i)13-s + (0.0602 − 0.0682i)15-s + (−2.88 + 4.99i)17-s + (−2.92 + 1.69i)19-s + (3.01 + 3.44i)21-s + (−7.47 − 4.31i)23-s + (2.49 + 4.32i)25-s + (−4.29 − 2.93i)27-s + (−6.23 − 3.60i)29-s + 9.92i·31-s + ⋯
L(s)  = 1  + (−0.980 − 0.198i)3-s + (−0.0117 + 0.0203i)5-s + (−0.795 − 0.606i)7-s + (0.921 + 0.389i)9-s + (0.648 − 0.374i)11-s + (−0.698 + 0.403i)13-s + (0.0155 − 0.0176i)15-s + (−0.700 + 1.21i)17-s + (−0.671 + 0.387i)19-s + (0.658 + 0.752i)21-s + (−1.55 − 0.899i)23-s + (0.499 + 0.865i)25-s + (−0.825 − 0.564i)27-s + (−1.15 − 0.668i)29-s + 1.78i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.838 - 0.544i$
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: \((2,\ 504,\ (\ :1/2),\ -0.838 - 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0424636 + 0.143498i\)
\(L(\frac12)\) \(\approx\) \(0.0424636 + 0.143498i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.343i)T \)
7 \( 1 + (2.10 + 1.60i)T \)
good5 \( 1 + (0.0262 - 0.0455i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.15 + 1.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.51 - 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.88 - 4.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.92 - 1.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.47 + 4.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.23 + 3.60i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.92iT - 31T^{2} \)
37 \( 1 + (0.770 + 1.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.392 + 0.679i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.03 - 3.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.31T + 47T^{2} \)
53 \( 1 + (0.710 + 0.410i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.64T + 59T^{2} \)
61 \( 1 + 5.62iT - 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 - 3.51iT - 71T^{2} \)
73 \( 1 + (-6.75 - 3.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + (-3.14 + 5.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.93 + 13.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.57 + 0.909i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.16374265957220437106028428101, −10.50482201871247180244125440147, −9.744769656424753744353544057617, −8.630271292689627153477260439594, −7.41448027375125678068625177883, −6.52575904313532901035548524190, −6.00308560167523273120056429350, −4.57057812962293199171276445505, −3.71635971524451236969172412194, −1.77915629298457530109475425184, 0.096081487176404460198390025090, 2.27972908064108423344272675950, 3.87332774221292251529753794697, 4.92592013449572498150084821803, 5.93441453022206171039608582167, 6.69649857171971246494746735477, 7.62935456066746971025582120920, 9.164925590668382943602629835027, 9.630557363243513336317186797885, 10.54707923138666685179666277739

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