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Properties

Label 2-546-13.12-c1-0-10
Degree $2$
Conductor $546$
Sign $-0.554 + 0.832i$
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  − i·2-s + 3-s − 4-s − 2i·5-s − i·6-s − i·7-s + i·8-s + 9-s − 2·10-s − 12-s + (2 − 3i)13-s − 14-s − 2i·15-s + 16-s − 2·17-s − i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.894i·5-s − 0.408i·6-s − 0.377i·7-s + 0.353i·8-s + 0.333·9-s − 0.632·10-s − 0.288·12-s + (0.554 − 0.832i)13-s − 0.267·14-s − 0.516i·15-s + 0.250·16-s − 0.485·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738680 - 1.38023i\)
\(L(\frac12)\) \(\approx\) \(0.738680 - 1.38023i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - T \)
7 \( 1 + iT \)
13 \( 1 + (-2 + 3i)T \)
good5 \( 1 + 2iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−10.43906599472386037674039697815, −9.659921115453137208029938945955, −8.691171084522185913450514798227, −8.235604395395760847355183186408, −7.02718194012533614405126100233, −5.62467068425307463596678292367, −4.56248524829342132890619276028, −3.66579191453345194167135106980, −2.36850151416646713137635551837, −0.893805411861998558286043851328, 2.05695015814643121629296445367, 3.43035668030050780103166388735, 4.40700276552644831119540956017, 5.87789916850495982186630237276, 6.57885924456970419944274634727, 7.51287583689047377352540681663, 8.369919440255692162891186696016, 9.189931348597325525389078539261, 10.06788693215347469742102062536, 10.98284679500704824209555115162

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