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Properties

Label 2-504-63.58-c1-0-5
Degree $2$
Conductor $504$
Sign $0.365 - 0.930i$
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  + (−1.56 − 0.746i)3-s + (1.71 + 2.97i)5-s + (−0.727 − 2.54i)7-s + (1.88 + 2.33i)9-s + (−2.20 + 3.81i)11-s + (1.49 − 2.58i)13-s + (−0.463 − 5.93i)15-s + (0.542 + 0.939i)17-s + (−3.74 + 6.48i)19-s + (−0.761 + 4.51i)21-s + (2.16 + 3.74i)23-s + (−3.40 + 5.89i)25-s + (−1.20 − 5.05i)27-s + (1.68 + 2.91i)29-s + 9.37·31-s + ⋯
L(s)  = 1  + (−0.902 − 0.431i)3-s + (0.768 + 1.33i)5-s + (−0.275 − 0.961i)7-s + (0.628 + 0.777i)9-s + (−0.664 + 1.15i)11-s + (0.414 − 0.717i)13-s + (−0.119 − 1.53i)15-s + (0.131 + 0.227i)17-s + (−0.858 + 1.48i)19-s + (−0.166 + 0.986i)21-s + (0.450 + 0.781i)23-s + (−0.680 + 1.17i)25-s + (−0.231 − 0.972i)27-s + (0.312 + 0.541i)29-s + 1.68·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.365 - 0.930i$
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: \((2,\ 504,\ (\ :1/2),\ 0.365 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849755 + 0.578916i\)
\(L(\frac12)\) \(\approx\) \(0.849755 + 0.578916i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 + 0.746i)T \)
7 \( 1 + (0.727 + 2.54i)T \)
good5 \( 1 + (-1.71 - 2.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.20 - 3.81i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.49 + 2.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.542 - 0.939i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.74 - 6.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.16 - 3.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.68 - 2.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.37T + 31T^{2} \)
37 \( 1 + (2.50 - 4.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.20 - 2.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.31 - 5.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.00T + 47T^{2} \)
53 \( 1 + (0.530 + 0.919i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 5.42T + 61T^{2} \)
67 \( 1 - 3.33T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (8.21 + 14.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 2.35T + 79T^{2} \)
83 \( 1 + (1.60 + 2.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.67 + 9.82i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.40 + 11.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.77811595330975823366792895918, −10.25405032790517774610953812924, −9.991005480684050075857577097625, −8.020488782992142795825475540574, −7.25181347650253512978229792817, −6.49032498987864570622360868623, −5.77845842129589045514236410212, −4.49465540177749325324995289912, −3.04529458586617551129552304256, −1.61791814322016252546307980054, 0.71349581990956818231571627342, 2.53700680035245966284451010718, 4.34042718495663379705871035007, 5.20516192160184072102437420656, 5.87835401822110712032079085117, 6.69545353782895949708199922325, 8.650242171894903690228603101639, 8.814315068218906322784271992920, 9.820770525172725737806544378231, 10.78960668121435131447451194988

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