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Properties

Label 2-504-9.4-c1-0-4
Degree $2$
Conductor $504$
Sign $-0.173 - 0.984i$
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  + 1.73i·3-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 2.99·9-s + (3 + 5.19i)11-s + (−3 + 5.19i)13-s + (1.49 + 0.866i)15-s − 2·17-s + 7·19-s + (1.49 − 0.866i)21-s + (0.5 − 0.866i)23-s + (2 + 3.46i)25-s − 5.19i·27-s + (−1 − 1.73i)29-s + (−5 + 8.66i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.223 − 0.387i)5-s + (−0.188 − 0.327i)7-s − 0.999·9-s + (0.904 + 1.56i)11-s + (−0.832 + 1.44i)13-s + (0.387 + 0.223i)15-s − 0.485·17-s + 1.60·19-s + (0.327 − 0.188i)21-s + (0.104 − 0.180i)23-s + (0.400 + 0.692i)25-s − 0.999i·27-s + (−0.185 − 0.321i)29-s + (−0.898 + 1.55i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840749 + 1.00196i\)
\(L(\frac12)\) \(\approx\) \(0.840749 + 1.00196i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.14378340108373810487657285953, −10.02993838421158318267995573610, −9.359254759688915349556621399809, −9.024934753800349899135184143228, −7.38053010498683180666534694982, −6.70237827658543844663769146392, −5.17418723791595700847497098426, −4.57478392809827290118619777348, −3.54373699626362443910998565605, −1.87888496616004382779534560991, 0.806745403398612268029706803062, 2.57894530970648102149339949371, 3.42933843704614823781603007959, 5.43647709877168708928633998530, 5.99138620537582391937242587918, 7.04079806487045421841667477210, 7.88063406999135904968873483830, 8.790442898467280386352089049275, 9.689927047430379837963445274130, 10.92216905095768275382057117270

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