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Properties

Label 2-504-56.37-c1-0-17
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: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27100 + 0.446832i\)
\(L(\frac12)\) \(\approx\) \(1.27100 + 0.446832i\)
\(L(\frac{3}{2})\) 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

−10.75990097534982516393583025181, −10.04807587690387444114025565587, −9.275304726630335605288066203302, −8.102065789825182746617596262616, −7.68469980203831102776674906764, −6.14350877334003932046109929852, −5.90014668132693045831516584828, −4.65316514243329645739887598894, −2.71591614145100259590656979226, −1.32629612755058273418819048084, 1.44444771847364729361040035116, 2.18003244141945899104708451941, 4.06264424092567810273248737205, 4.87704118197962221733598804120, 6.35805225996209393780129815464, 7.35238709022440259354881320796, 8.482095986924576513711156087445, 9.196471678127356703672493120967, 9.750803431930135572047778906297, 10.83913190143468419490409388545

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