Properties

Label 2-504-7.4-c1-0-6
Degree $2$
Conductor $504$
Sign $0.605 + 0.795i$
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  + (−0.5 − 0.866i)5-s + (2 − 1.73i)7-s + (−0.5 + 0.866i)11-s + 2·13-s + (1.5 − 2.59i)17-s + (−2.5 − 4.33i)19-s + (−1.5 − 2.59i)23-s + (2 − 3.46i)25-s + 6·29-s + (0.5 − 0.866i)31-s + (−2.5 − 0.866i)35-s + (2.5 + 4.33i)37-s + 10·41-s − 4·43-s + (0.5 + 0.866i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.755 − 0.654i)7-s + (−0.150 + 0.261i)11-s + 0.554·13-s + (0.363 − 0.630i)17-s + (−0.573 − 0.993i)19-s + (−0.312 − 0.541i)23-s + (0.400 − 0.692i)25-s + 1.11·29-s + (0.0898 − 0.155i)31-s + (−0.422 − 0.146i)35-s + (0.410 + 0.711i)37-s + 1.56·41-s − 0.609·43-s + (0.0729 + 0.126i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30303 - 0.645948i\)
\(L(\frac12)\) \(\approx\) \(1.30303 - 0.645948i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−10.81627764745146234250609139333, −10.02052788068139029367019706633, −8.853029998178216452447766299709, −8.145060202979030318588492369731, −7.23916708911857770745153017751, −6.21262422190085422810323435234, −4.82964162963559734207292148483, −4.27717280033703008354362069211, −2.67923754608332172309319441548, −0.979496238308350598750057510035, 1.66627532062562487822964824400, 3.11955579412677930084823981767, 4.28961735523182977106176245781, 5.55268518718739421481797042753, 6.29964388284551260373170734802, 7.63346575549667969463888815211, 8.288317764500899715875536372954, 9.157884635645311769706777384924, 10.36810583706330259437299516461, 11.00586689573546524503265085121

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