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Properties

Label 2-504-56.13-c2-0-33
Degree $2$
Conductor $504$
Sign $0.265 - 0.964i$
Analytic cond. $13.7330$
Root an. cond. $3.70580$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.84 + 0.762i)2-s + (2.83 + 2.81i)4-s + 2.18·5-s + (−5.50 − 4.32i)7-s + (3.09 + 7.37i)8-s + (4.04 + 1.66i)10-s + 13.7i·11-s + 7.07·13-s + (−6.87 − 12.1i)14-s + (0.0962 + 15.9i)16-s + 27.5i·17-s + 30.2·19-s + (6.20 + 6.16i)20-s + (−10.5 + 25.4i)22-s + 29.0·23-s + ⋯
L(s)  = 1  + (0.924 + 0.381i)2-s + (0.709 + 0.704i)4-s + 0.437·5-s + (−0.786 − 0.618i)7-s + (0.386 + 0.922i)8-s + (0.404 + 0.166i)10-s + 1.25i·11-s + 0.544·13-s + (−0.491 − 0.871i)14-s + (0.00601 + 0.999i)16-s + 1.62i·17-s + 1.59·19-s + (0.310 + 0.308i)20-s + (−0.478 + 1.15i)22-s + 1.26·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.265 - 0.964i$
Analytic conductor: \(13.7330\)
Root analytic conductor: \(3.70580\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1),\ 0.265 - 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.181924513\)
\(L(\frac12)\) \(\approx\) \(3.181924513\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.84 - 0.762i)T \)
3 \( 1 \)
7 \( 1 + (5.50 + 4.32i)T \)
good5 \( 1 - 2.18T + 25T^{2} \)
11 \( 1 - 13.7iT - 121T^{2} \)
13 \( 1 - 7.07T + 169T^{2} \)
17 \( 1 - 27.5iT - 289T^{2} \)
19 \( 1 - 30.2T + 361T^{2} \)
23 \( 1 - 29.0T + 529T^{2} \)
29 \( 1 - 21.2iT - 841T^{2} \)
31 \( 1 + 45.7iT - 961T^{2} \)
37 \( 1 - 41.0iT - 1.36e3T^{2} \)
41 \( 1 + 60.4iT - 1.68e3T^{2} \)
43 \( 1 + 33.6iT - 1.84e3T^{2} \)
47 \( 1 + 9.53iT - 2.20e3T^{2} \)
53 \( 1 + 46.9iT - 2.80e3T^{2} \)
59 \( 1 + 95.6T + 3.48e3T^{2} \)
61 \( 1 + 45.4T + 3.72e3T^{2} \)
67 \( 1 + 19.4iT - 4.48e3T^{2} \)
71 \( 1 + 4.84T + 5.04e3T^{2} \)
73 \( 1 - 5.02iT - 5.32e3T^{2} \)
79 \( 1 - 110.T + 6.24e3T^{2} \)
83 \( 1 - 17.9T + 6.88e3T^{2} \)
89 \( 1 + 10.6iT - 7.92e3T^{2} \)
97 \( 1 - 8.67iT - 9.40e3T^{2} \)
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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

−10.91893197079707619793473040809, −10.10849186484510655630273122324, −9.177953213384447263103098203117, −7.84497100331744984502301989764, −7.07274897012399021931246549708, −6.24094311903312315031871751159, −5.30678050787856300278668972604, −4.12640770641208945932475681800, −3.24913789036361751044289113477, −1.73478634207551566399329354040, 1.01440864888082616513018834139, 2.82088318551848595700071564991, 3.32181472559922168172822832292, 4.96045032823265874501603676909, 5.74457511095359728188037869756, 6.48407036394496885537595072084, 7.59283249008038753538398291495, 9.153829133365995403859170827320, 9.573635435328651332251860738518, 10.78498748855977372923636243349

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