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Properties

Label 2-546-91.9-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.552 + 0.833i$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (1.75 − 3.03i)5-s + (−0.5 − 0.866i)6-s + (1.12 − 2.39i)7-s + 0.999·8-s + 9-s − 3.50·10-s − 6.40·11-s + (−0.499 + 0.866i)12-s + (−0.213 − 3.59i)13-s + (−2.63 + 0.222i)14-s + (1.75 − 3.03i)15-s + (−0.5 − 0.866i)16-s + (−2.33 + 4.05i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + 0.577·3-s + (−0.249 + 0.433i)4-s + (0.784 − 1.35i)5-s + (−0.204 − 0.353i)6-s + (0.425 − 0.904i)7-s + 0.353·8-s + 0.333·9-s − 1.10·10-s − 1.93·11-s + (−0.144 + 0.249i)12-s + (−0.0590 − 0.998i)13-s + (−0.704 + 0.0593i)14-s + (0.452 − 0.784i)15-s + (−0.125 − 0.216i)16-s + (−0.567 + 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.552 + 0.833i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.552 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.707572 - 1.31779i\)
\(L(\frac12)\) \(\approx\) \(0.707572 - 1.31779i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 - T \)
7 \( 1 + (-1.12 + 2.39i)T \)
13 \( 1 + (0.213 + 3.59i)T \)
good5 \( 1 + (-1.75 + 3.03i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 6.40T + 11T^{2} \)
17 \( 1 + (2.33 - 4.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 5.23T + 19T^{2} \)
23 \( 1 + (-1.08 - 1.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.46 - 7.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.94 + 3.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.09 + 8.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.19 + 2.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.44 - 4.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.05 + 1.82i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.89 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 9.35T + 61T^{2} \)
67 \( 1 - 7.28T + 67T^{2} \)
71 \( 1 + (-2.79 - 4.83i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.23 - 7.32i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.893 + 1.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.59T + 83T^{2} \)
89 \( 1 + (3.50 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.92 + 8.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.31200421833015009668915640300, −9.796027903880495445546817439314, −8.638862744578295599883648477131, −8.148265912792385311411488261024, −7.30135313410037407067810010439, −5.42151480551124246180278407083, −4.91309906125712942382672605421, −3.51782077659365758640795794174, −2.18961568760642490783627567445, −0.930571167881182337561271294132, 2.28231594433722869996505642743, 2.81517582259553206086815275840, 4.79120631166993845307389302396, 5.71090409163676258884132094811, 6.70089030768702016370080643283, 7.54591307169111574813195174408, 8.307921297696259333121799329856, 9.506599098114992844521719359351, 9.879739262787791164735823697196, 10.96451858502411805724033049375

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