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Properties

Label 2-504-9.4-c1-0-15
Degree $2$
Conductor $504$
Sign $-0.271 + 0.962i$
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  + (0.385 − 1.68i)3-s + (1.15 − 2.00i)5-s + (0.5 + 0.866i)7-s + (−2.70 − 1.30i)9-s + (−0.655 − 1.13i)11-s + (1.93 − 3.34i)13-s + (−2.93 − 2.72i)15-s − 0.326·17-s + 3.08·19-s + (1.65 − 0.510i)21-s + (−1.81 + 3.15i)23-s + (−0.169 − 0.293i)25-s + (−3.24 + 4.05i)27-s + (−4.75 − 8.22i)29-s + (−3.24 + 5.62i)31-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)3-s + (0.516 − 0.894i)5-s + (0.188 + 0.327i)7-s + (−0.900 − 0.434i)9-s + (−0.197 − 0.342i)11-s + (0.536 − 0.928i)13-s + (−0.757 − 0.703i)15-s − 0.0792·17-s + 0.707·19-s + (0.361 − 0.111i)21-s + (−0.379 + 0.656i)23-s + (−0.0338 − 0.0586i)25-s + (−0.624 + 0.781i)27-s + (−0.882 − 1.52i)29-s + (−0.583 + 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.271 + 0.962i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.958582 - 1.26627i\)
\(L(\frac12)\) \(\approx\) \(0.958582 - 1.26627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.385 + 1.68i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.15 + 2.00i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.655 + 1.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.93 + 3.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.326T + 17T^{2} \)
19 \( 1 - 3.08T + 19T^{2} \)
23 \( 1 + (1.81 - 3.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.75 + 8.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.24 - 5.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.31T + 37T^{2} \)
41 \( 1 + (-4.74 + 8.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0493 + 0.0855i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.108 + 0.187i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.7T + 53T^{2} \)
59 \( 1 + (-1.22 + 2.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.68 - 13.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.93 + 5.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.77T + 71T^{2} \)
73 \( 1 + 5.99T + 73T^{2} \)
79 \( 1 + (-7.29 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.04 - 5.26i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.52T + 89T^{2} \)
97 \( 1 + (-2.99 - 5.18i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.75516872369746831575962288655, −9.526078282879790712178879925910, −8.756605409928610108627105545690, −8.042442973879697432597479854894, −7.13059500443636853922142728093, −5.69099164081745093856222283056, −5.48658676957925244948803876295, −3.64155005628652052368719908586, −2.26157388455282155962810018168, −0.976496652864262830075791129622, 2.14755539125953380655279865389, 3.38276684309704965460566045027, 4.38466951250322450850571944767, 5.51519077007239178872736516864, 6.56056795049916382074107697439, 7.54505000914826765377130965738, 8.725365028694619667323640688607, 9.540122521458873562680293933800, 10.30515082697780745632081293766, 10.98564496336528116864775875350

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