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Properties

Label 2-504-63.59-c1-0-22
Degree $2$
Conductor $504$
Sign $-0.882 - 0.471i$
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  + (−1.45 − 0.938i)3-s − 1.81·5-s + (1.69 − 2.03i)7-s + (1.23 + 2.73i)9-s + 0.255i·11-s + (−5.77 − 3.33i)13-s + (2.63 + 1.69i)15-s + (−1.99 + 3.46i)17-s + (1.24 − 0.719i)19-s + (−4.37 + 1.37i)21-s + 5.66i·23-s − 1.71·25-s + (0.758 − 5.14i)27-s + (−4.18 + 2.41i)29-s + (−8.80 + 5.08i)31-s + ⋯
L(s)  = 1  + (−0.840 − 0.541i)3-s − 0.810·5-s + (0.639 − 0.768i)7-s + (0.413 + 0.910i)9-s + 0.0771i·11-s + (−1.60 − 0.925i)13-s + (0.681 + 0.438i)15-s + (−0.484 + 0.839i)17-s + (0.286 − 0.165i)19-s + (−0.954 + 0.299i)21-s + 1.18i·23-s − 0.343·25-s + (0.145 − 0.989i)27-s + (−0.777 + 0.448i)29-s + (−1.58 + 0.913i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.882 - 0.471i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.882 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0143115 + 0.0571711i\)
\(L(\frac12)\) \(\approx\) \(0.0143115 + 0.0571711i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 + 0.938i)T \)
7 \( 1 + (-1.69 + 2.03i)T \)
good5 \( 1 + 1.81T + 5T^{2} \)
11 \( 1 - 0.255iT - 11T^{2} \)
13 \( 1 + (5.77 + 3.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.99 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.24 + 0.719i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.66iT - 23T^{2} \)
29 \( 1 + (4.18 - 2.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.80 - 5.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.65 + 2.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.10 - 8.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.12 - 1.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.97 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.97 + 2.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.55 + 4.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.60 - 4.96i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.962 + 1.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.31iT - 71T^{2} \)
73 \( 1 + (2.47 + 1.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.83 + 3.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.68 - 4.64i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.378 + 0.655i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.21 + 2.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.66785703697306262281899458092, −9.753225901243440656400928498141, −8.259166695657216387701993144476, −7.45178213083399381197637166028, −7.08995225143577078694679708664, −5.54853838344418114029747110679, −4.82506142134895832748476397193, −3.62930109254275480976883505174, −1.76861266997825230043113519679, −0.03760821169261101950983649651, 2.30903471165339250500837300781, 4.01236077447891431014897734499, 4.80270103080156056264819937511, 5.64914096475091692959821330448, 6.93096866983042033324558114553, 7.69295035855497892180393435799, 9.000980241470634058984411648290, 9.579566920338484397337666810883, 10.75973436704716831063643232904, 11.56601588953178888195515645972

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