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Properties

Label 2-546-91.80-c1-0-13
Degree $2$
Conductor $546$
Sign $-0.457 + 0.889i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  + (−0.258 − 0.965i)2-s + i·3-s + (−0.866 + 0.499i)4-s + (0.782 − 2.92i)5-s + (0.965 − 0.258i)6-s + (1.58 − 2.12i)7-s + (0.707 + 0.707i)8-s − 9-s − 3.02·10-s + (−1.55 − 1.55i)11-s + (−0.499 − 0.866i)12-s + (1.15 + 3.41i)13-s + (−2.45 − 0.977i)14-s + (2.92 + 0.782i)15-s + (0.500 − 0.866i)16-s + (−2.40 − 4.15i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + 0.577i·3-s + (−0.433 + 0.249i)4-s + (0.349 − 1.30i)5-s + (0.394 − 0.105i)6-s + (0.597 − 0.801i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s − 0.956·10-s + (−0.469 − 0.469i)11-s + (−0.144 − 0.249i)12-s + (0.319 + 0.947i)13-s + (−0.657 − 0.261i)14-s + (0.754 + 0.202i)15-s + (0.125 − 0.216i)16-s + (−0.582 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.457 + 0.889i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.457 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.636347 - 1.04349i\)
\(L(\frac12)\) \(\approx\) \(0.636347 - 1.04349i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 - iT \)
7 \( 1 + (-1.58 + 2.12i)T \)
13 \( 1 + (-1.15 - 3.41i)T \)
good5 \( 1 + (-0.782 + 2.92i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.55 + 1.55i)T + 11iT^{2} \)
17 \( 1 + (2.40 + 4.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.01 + 2.01i)T + 19iT^{2} \)
23 \( 1 + (0.721 + 0.416i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.310 - 0.538i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.04 + 0.547i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.98 + 1.06i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.82 + 10.5i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.69 + 3.28i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.82 - 1.02i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.61 + 9.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.61 + 2.03i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 7.60iT - 61T^{2} \)
67 \( 1 + (5.74 - 5.74i)T - 67iT^{2} \)
71 \( 1 + (-2.25 - 8.42i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.17 - 15.5i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.859 - 1.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.5 - 12.5i)T + 83iT^{2} \)
89 \( 1 + (-1.91 - 7.15i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-11.7 + 3.14i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.57353122377061627352461408388, −9.598177902649658583924548500735, −8.876173780323549236977846575852, −8.293858571979558336784887099926, −6.99747570743713902619402938087, −5.45063938277547667545134661714, −4.65650155163438766049245206565, −3.94609184934572112559005793019, −2.25290632399791107820493685633, −0.76938299441377568005771217321, 1.92086041713149482841574604489, 3.06203322602245458375744916230, 4.74811448818012305456438148645, 6.10103913453137689104305803863, 6.28176291356835042031083959578, 7.66166540444287374247153563227, 8.072838555003358344681967842143, 9.168370509441716658668943711326, 10.37602152247679049645717416049, 10.81604557682574242933308909077

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