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Properties

Label 2-504-56.37-c1-0-7
Degree $2$
Conductor $504$
Sign $-0.850 - 0.525i$
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  + (−0.491 + 1.32i)2-s + (−1.51 − 1.30i)4-s + (3.08 + 1.78i)5-s + (−2.38 + 1.14i)7-s + (2.47 − 1.36i)8-s + (−3.88 + 3.21i)10-s + (−3.52 + 2.03i)11-s − 1.44i·13-s + (−0.350 − 3.72i)14-s + (0.595 + 3.95i)16-s + (3.49 + 6.05i)17-s + (0.261 + 0.150i)19-s + (−2.35 − 6.73i)20-s + (−0.964 − 5.67i)22-s + (−1.21 + 2.10i)23-s + ⋯
L(s)  = 1  + (−0.347 + 0.937i)2-s + (−0.757 − 0.652i)4-s + (1.38 + 0.797i)5-s + (−0.900 + 0.434i)7-s + (0.875 − 0.483i)8-s + (−1.22 + 1.01i)10-s + (−1.06 + 0.613i)11-s − 0.399i·13-s + (−0.0936 − 0.995i)14-s + (0.148 + 0.988i)16-s + (0.847 + 1.46i)17-s + (0.0599 + 0.0345i)19-s + (−0.526 − 1.50i)20-s + (−0.205 − 1.21i)22-s + (−0.252 + 0.437i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.293183 + 1.03226i\)
\(L(\frac12)\) \(\approx\) \(0.293183 + 1.03226i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.491 - 1.32i)T \)
3 \( 1 \)
7 \( 1 + (2.38 - 1.14i)T \)
good5 \( 1 + (-3.08 - 1.78i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.52 - 2.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.44iT - 13T^{2} \)
17 \( 1 + (-3.49 - 6.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.261 - 0.150i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.21 - 2.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.151iT - 29T^{2} \)
31 \( 1 + (-2.37 - 4.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.82 + 5.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.239T + 41T^{2} \)
43 \( 1 - 1.32iT - 43T^{2} \)
47 \( 1 + (3.17 - 5.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.18 - 2.99i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.73 + 5.62i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.64 - 2.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.79 + 2.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + (0.284 + 0.493i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.746 + 1.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 + (1.83 - 3.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−10.68848788354538147416754236173, −10.13634526932888477125561407469, −9.684967225325812818435623924918, −8.590870512448081242823567101535, −7.55118202691182474402256815021, −6.58263215930256728210496520227, −5.87255370569298741262288476706, −5.22047955982183103943860275946, −3.37930464553749081985529281597, −1.96702430068027300071649566137, 0.71556866075823707008535095281, 2.27415471493277251305482713246, 3.32473290626045280820029310451, 4.85288486155740801777925694437, 5.61332531914133569130852634755, 6.94107155695130318378773192322, 8.202288404788546489381695002373, 9.064110669806963677297488388732, 9.974079181653080870833122972368, 10.10279383025120914322393685413

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