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Properties

Label 2-546-13.12-c7-0-68
Degree $2$
Conductor $546$
Sign $0.907 + 0.420i$
Analytic cond. $170.562$
Root an. cond. $13.0599$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8i·2-s − 27·3-s − 64·4-s + 208. i·5-s − 216i·6-s − 343i·7-s − 512i·8-s + 729·9-s − 1.66e3·10-s − 1.97e3i·11-s + 1.72e3·12-s + (7.18e3 + 3.32e3i)13-s + 2.74e3·14-s − 5.63e3i·15-s + 4.09e3·16-s + 727.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.746i·5-s − 0.408i·6-s − 0.377i·7-s − 0.353i·8-s + 0.333·9-s − 0.527·10-s − 0.448i·11-s + 0.288·12-s + (0.907 + 0.420i)13-s + 0.267·14-s − 0.431i·15-s + 0.250·16-s + 0.0358·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.907 + 0.420i$
Analytic conductor: \(170.562\)
Root analytic conductor: \(13.0599\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :7/2),\ 0.907 + 0.420i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.147951284\)
\(L(\frac12)\) \(\approx\) \(1.147951284\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8iT \)
3 \( 1 + 27T \)
7 \( 1 + 343iT \)
13 \( 1 + (-7.18e3 - 3.32e3i)T \)
good5 \( 1 - 208. iT - 7.81e4T^{2} \)
11 \( 1 + 1.97e3iT - 1.94e7T^{2} \)
17 \( 1 - 727.T + 4.10e8T^{2} \)
19 \( 1 + 1.77e3iT - 8.93e8T^{2} \)
23 \( 1 - 6.78e4T + 3.40e9T^{2} \)
29 \( 1 + 1.87e5T + 1.72e10T^{2} \)
31 \( 1 + 4.59e4iT - 2.75e10T^{2} \)
37 \( 1 + 5.67e4iT - 9.49e10T^{2} \)
41 \( 1 - 4.39e5iT - 1.94e11T^{2} \)
43 \( 1 + 1.02e5T + 2.71e11T^{2} \)
47 \( 1 + 1.32e6iT - 5.06e11T^{2} \)
53 \( 1 + 1.68e5T + 1.17e12T^{2} \)
59 \( 1 - 2.27e6iT - 2.48e12T^{2} \)
61 \( 1 + 1.95e6T + 3.14e12T^{2} \)
67 \( 1 - 2.89e5iT - 6.06e12T^{2} \)
71 \( 1 + 5.13e6iT - 9.09e12T^{2} \)
73 \( 1 + 4.29e6iT - 1.10e13T^{2} \)
79 \( 1 + 2.00e6T + 1.92e13T^{2} \)
83 \( 1 - 2.06e4iT - 2.71e13T^{2} \)
89 \( 1 + 2.46e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.30e7iT - 8.07e13T^{2} \)
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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

−9.520790712194631007586423349253, −8.671796087468621229353968279068, −7.55894350178779372142633645574, −6.82130461954578520807917617102, −6.12721476005698745518719812705, −5.18982710711529153347519718416, −4.06198684730407092579161578114, −3.11394478308521751855920564182, −1.46150874769079219862945665045, −0.29863836074932462279257397425, 0.876631966595421729759003975627, 1.64458594759626149508731725482, 2.99966497176035238491857902976, 4.13975514091598510550868586100, 5.08685522843485185203950619134, 5.78313601404916812408448224118, 7.00837666801654491536722919211, 8.188535123668872550590663275498, 9.021537499945702965012066865161, 9.716602699083157471762221222295

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