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Properties

Label 2-504-56.53-c1-0-10
Degree $2$
Conductor $504$
Sign $-0.780 - 0.625i$
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  + (1.01 + 0.987i)2-s + (0.0488 + 1.99i)4-s + (−2.26 + 1.30i)5-s + (2.62 − 0.358i)7-s + (−1.92 + 2.07i)8-s + (−3.58 − 0.912i)10-s + (−1.87 − 1.08i)11-s + 6.50i·13-s + (3.00 + 2.22i)14-s + (−3.99 + 0.195i)16-s + (−0.797 + 1.38i)17-s + (−1.27 + 0.736i)19-s + (−2.72 − 4.46i)20-s + (−0.828 − 2.94i)22-s + (1.92 + 3.33i)23-s + ⋯
L(s)  = 1  + (0.715 + 0.698i)2-s + (0.0244 + 0.999i)4-s + (−1.01 + 0.584i)5-s + (0.990 − 0.135i)7-s + (−0.680 + 0.732i)8-s + (−1.13 − 0.288i)10-s + (−0.565 − 0.326i)11-s + 1.80i·13-s + (0.803 + 0.594i)14-s + (−0.998 + 0.0488i)16-s + (−0.193 + 0.335i)17-s + (−0.292 + 0.169i)19-s + (−0.608 − 0.997i)20-s + (−0.176 − 0.628i)22-s + (0.401 + 0.695i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.541510 + 1.54030i\)
\(L(\frac12)\) \(\approx\) \(0.541510 + 1.54030i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.01 - 0.987i)T \)
3 \( 1 \)
7 \( 1 + (-2.62 + 0.358i)T \)
good5 \( 1 + (2.26 - 1.30i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.87 + 1.08i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.50iT - 13T^{2} \)
17 \( 1 + (0.797 - 1.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.27 - 0.736i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.92 - 3.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.39iT - 29T^{2} \)
31 \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.69 + 3.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.70T + 41T^{2} \)
43 \( 1 + 2.69iT - 43T^{2} \)
47 \( 1 + (-5.77 - 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.66 - 5.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.11 - 4.68i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.60 - 2.08i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.99 - 5.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.62 + 6.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.20iT - 83T^{2} \)
89 \( 1 + (3.85 + 6.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.828T + 97T^{2} \)
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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

−11.61791998734646292810818513963, −10.76302422175595414444621111890, −9.207092578222270046091304702251, −8.202985637196750755098109560837, −7.58335759066862512612151886017, −6.79128899669052142835259521590, −5.68756069301253609254781790168, −4.40915048049777299947926038486, −3.90792813044742194243239065352, −2.36536532969004961651361066328, 0.795193486975663015833426371400, 2.53020456016315852135223231442, 3.74183500880790079999624090224, 4.96183877114554783336262638383, 5.24252784341917200608672428492, 6.88027700291791945266811644691, 8.072892086324642547996600118299, 8.610163583953044783174648728416, 10.04610466374734393785440558798, 10.78656868968041578191158406913

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