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Properties

Label 2-504-504.293-c1-0-50
Degree $2$
Conductor $504$
Sign $0.617 + 0.786i$
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  + (1.22 − 0.707i)2-s + (−1.68 + 0.420i)3-s + (0.999 − 1.73i)4-s + (−0.0658 − 0.0380i)5-s + (−1.76 + 1.70i)6-s + (1.32 + 2.29i)7-s − 2.82i·8-s + (2.64 − 1.41i)9-s − 0.107·10-s + (−0.951 + 3.33i)12-s + (2.27 − 3.94i)13-s + (3.24 + 1.87i)14-s + (0.126 + 0.0361i)15-s + (−2.00 − 3.46i)16-s + (2.24 − 3.60i)18-s + 7.99·19-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (−0.970 + 0.242i)3-s + (0.499 − 0.866i)4-s + (−0.0294 − 0.0169i)5-s + (−0.718 + 0.695i)6-s + (0.499 + 0.866i)7-s − 0.999i·8-s + (0.881 − 0.471i)9-s − 0.0339·10-s + (−0.274 + 0.961i)12-s + (0.631 − 1.09i)13-s + (0.866 + 0.499i)14-s + (0.0326 + 0.00933i)15-s + (−0.500 − 0.866i)16-s + (0.528 − 0.849i)18-s + 1.83·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71353 - 0.833424i\)
\(L(\frac12)\) \(\approx\) \(1.71353 - 0.833424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (1.68 - 0.420i)T \)
7 \( 1 + (-1.32 - 2.29i)T \)
good5 \( 1 + (0.0658 + 0.0380i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.27 + 3.94i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.99T + 19T^{2} \)
23 \( 1 + (-1.25 - 0.727i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (4.58 + 2.64i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.13 - 7.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (-6.02 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (15.7 - 9.08i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 - 84.0i)T^{2} \)
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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

−11.07361277479170397937359667292, −10.17871542977555533230100041318, −9.390744257219719822570736119590, −7.988401864578934227632610902825, −6.77460265880136259063473800628, −5.56529383592173662975099402262, −5.41219194174688941281636589243, −4.12026568366258612631637547306, −2.88926731272465208471697641122, −1.19282236489269154997024931278, 1.55279679069427808171358475570, 3.55286704281813747772240602149, 4.56098994263317388222383203037, 5.38205858778266639462605176231, 6.40512052205581551949747267105, 7.23410368158981596881015316954, 7.83233152697623410550178570692, 9.261187808690802992467701441805, 10.52093174383051797455837297661, 11.46709278852143128267199588895

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