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Properties

Label 2-504-63.58-c1-0-14
Degree $2$
Conductor $504$
Sign $0.994 + 0.106i$
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  + (1.34 − 1.09i)3-s + (0.918 + 1.59i)5-s + (−0.361 + 2.62i)7-s + (0.616 − 2.93i)9-s + (1.54 − 2.68i)11-s + (2.40 − 4.16i)13-s + (2.97 + 1.13i)15-s + (1.87 + 3.24i)17-s + (−2.71 + 4.70i)19-s + (2.37 + 3.91i)21-s + (3.97 + 6.89i)23-s + (0.813 − 1.40i)25-s + (−2.37 − 4.62i)27-s + (−0.325 − 0.563i)29-s + 1.03·31-s + ⋯
L(s)  = 1  + (0.776 − 0.630i)3-s + (0.410 + 0.711i)5-s + (−0.136 + 0.990i)7-s + (0.205 − 0.978i)9-s + (0.466 − 0.808i)11-s + (0.666 − 1.15i)13-s + (0.767 + 0.293i)15-s + (0.453 + 0.786i)17-s + (−0.622 + 1.07i)19-s + (0.518 + 0.855i)21-s + (0.829 + 1.43i)23-s + (0.162 − 0.281i)25-s + (−0.457 − 0.889i)27-s + (−0.0604 − 0.104i)29-s + 0.186·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.994 + 0.106i$
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: \((2,\ 504,\ (\ :1/2),\ 0.994 + 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01771 - 0.107346i\)
\(L(\frac12)\) \(\approx\) \(2.01771 - 0.107346i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 + 1.09i)T \)
7 \( 1 + (0.361 - 2.62i)T \)
good5 \( 1 + (-0.918 - 1.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.54 + 2.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.87 - 3.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.71 - 4.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.97 - 6.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.325 + 0.563i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.03T + 31T^{2} \)
37 \( 1 + (-0.873 + 1.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.52 + 4.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.09 + 10.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.61T + 47T^{2} \)
53 \( 1 + (-4.55 - 7.88i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.79T + 59T^{2} \)
61 \( 1 + 4.81T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + 5.00T + 71T^{2} \)
73 \( 1 + (1.81 + 3.14i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 + (-3.83 - 6.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.76 - 9.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.04 + 1.80i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.82242180913176404901724510008, −9.990594560379944037847153755870, −8.881213915100318995238698708716, −8.355732303984146097525964510756, −7.34943750994599353135189860456, −6.09334738115151862348987190817, −5.79693239278800586204965797093, −3.62233668771464228107812519775, −2.93518525114716716931953794159, −1.58574435137824823883655947830, 1.48998980260642113888603623146, 3.02408217996056417416742578612, 4.50522940650914346467010791825, 4.66592261375765874916022148931, 6.50717710496610956592809265808, 7.30359460736105639419452449864, 8.537967636667796243132288830113, 9.205029041159290137328551231054, 9.835558916745931259542356844974, 10.78291249810252197315531425848

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