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Properties

Label 2-546-91.83-c1-0-13
Degree $2$
Conductor $546$
Sign $-0.970 + 0.241i$
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  + (0.707 − 0.707i)2-s + i·3-s − 1.00i·4-s + (−2.56 − 2.56i)5-s + (0.707 + 0.707i)6-s + (−0.648 + 2.56i)7-s + (−0.707 − 0.707i)8-s − 9-s − 3.62·10-s + (−1.64 − 1.64i)11-s + 1.00·12-s + (2 − 3i)13-s + (1.35 + 2.27i)14-s + (2.56 − 2.56i)15-s − 1.00·16-s − 7.15·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + 0.577i·3-s − 0.500i·4-s + (−1.14 − 1.14i)5-s + (0.288 + 0.288i)6-s + (−0.245 + 0.969i)7-s + (−0.250 − 0.250i)8-s − 0.333·9-s − 1.14·10-s + (−0.496 − 0.496i)11-s + 0.288·12-s + (0.554 − 0.832i)13-s + (0.362 + 0.607i)14-s + (0.662 − 0.662i)15-s − 0.250·16-s − 1.73·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.970 + 0.241i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0772648 - 0.630570i\)
\(L(\frac12)\) \(\approx\) \(0.0772648 - 0.630570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 - iT \)
7 \( 1 + (0.648 - 2.56i)T \)
13 \( 1 + (-2 + 3i)T \)
good5 \( 1 + (2.56 + 2.56i)T + 5iT^{2} \)
11 \( 1 + (1.64 + 1.64i)T + 11iT^{2} \)
17 \( 1 + 7.15T + 17T^{2} \)
19 \( 1 + (2.86 + 2.86i)T + 19iT^{2} \)
23 \( 1 + 4.45iT - 23T^{2} \)
29 \( 1 + 9.75T + 29T^{2} \)
31 \( 1 + (-3.91 - 3.91i)T + 31iT^{2} \)
37 \( 1 + (-2.48 - 2.48i)T + 37iT^{2} \)
41 \( 1 + (-7.53 - 7.53i)T + 41iT^{2} \)
43 \( 1 + 8.62iT - 43T^{2} \)
47 \( 1 + (0.703 - 0.703i)T - 47iT^{2} \)
53 \( 1 - 2.42T + 53T^{2} \)
59 \( 1 + (-10.4 + 10.4i)T - 59iT^{2} \)
61 \( 1 + 7.15iT - 61T^{2} \)
67 \( 1 + (4.53 - 4.53i)T - 67iT^{2} \)
71 \( 1 + (0.456 - 0.456i)T - 71iT^{2} \)
73 \( 1 + (-2.48 + 2.48i)T - 73iT^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + (-2.70 - 2.70i)T + 83iT^{2} \)
89 \( 1 + (-4.75 + 4.75i)T - 89iT^{2} \)
97 \( 1 + (4.49 + 4.49i)T + 97iT^{2} \)
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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

−10.77726129365183828466764318729, −9.368727446223893383001890027994, −8.684935787456966673164657476810, −8.135548071029038766292602601475, −6.46921316127940983806029536065, −5.36455446516482753809464432421, −4.63006749680732696696892145215, −3.71350996810764682342609927592, −2.49878100154280320485867008824, −0.29433393339490897134034674654, 2.34419270266775474645925308968, 3.80129379115286667880165489356, 4.25755149935077928625378617282, 6.02445429643076634150397876167, 6.88204292377276054811121995163, 7.37166774323624470122841615208, 8.076531650467423906083189946030, 9.334370111164081602765638104915, 10.77053930353413624315575919759, 11.16070722632165814711016183993

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