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Properties

Label 2-546-91.54-c1-0-2
Degree $2$
Conductor $546$
Sign $0.0732 - 0.997i$
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 + (−0.866 − 0.5i)3-s − 1.00i·4-s + (−1.42 + 0.382i)5-s + (−0.965 + 0.258i)6-s + (0.349 + 2.62i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (−0.739 + 1.28i)10-s + (−5.63 + 1.51i)11-s + (−0.500 + 0.866i)12-s + (−2.68 + 2.41i)13-s + (2.10 + 1.60i)14-s + (1.42 + 0.382i)15-s − 1.00·16-s + 6.94·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.499 − 0.288i)3-s − 0.500i·4-s + (−0.638 + 0.171i)5-s + (−0.394 + 0.105i)6-s + (0.132 + 0.991i)7-s + (−0.250 − 0.250i)8-s + (0.166 + 0.288i)9-s + (−0.233 + 0.404i)10-s + (−1.69 + 0.455i)11-s + (−0.144 + 0.250i)12-s + (−0.743 + 0.668i)13-s + (0.561 + 0.429i)14-s + (0.368 + 0.0987i)15-s − 0.250·16-s + 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0732 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0732 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0732 - 0.997i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0732 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506337 + 0.470503i\)
\(L(\frac12)\) \(\approx\) \(0.506337 + 0.470503i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.349 - 2.62i)T \)
13 \( 1 + (2.68 - 2.41i)T \)
good5 \( 1 + (1.42 - 0.382i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.63 - 1.51i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 6.94T + 17T^{2} \)
19 \( 1 + (1.37 - 5.14i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 0.769iT - 23T^{2} \)
29 \( 1 + (4.35 + 7.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.101 - 0.379i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-4.14 - 4.14i)T + 37iT^{2} \)
41 \( 1 + (1.70 - 6.34i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.142 - 0.0825i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.23 - 12.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.44 + 7.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.616 + 0.616i)T - 59iT^{2} \)
61 \( 1 + (-4.89 + 2.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.23 + 4.61i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.36 + 5.08i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (8.94 + 2.39i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.96 - 6.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.29 + 9.29i)T + 83iT^{2} \)
89 \( 1 + (-6.30 + 6.30i)T - 89iT^{2} \)
97 \( 1 + (-0.852 + 0.228i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.32055778223548629949246021114, −10.17347757854929565866066633109, −9.647990425245496529950953363435, −7.967431801079099118849184873461, −7.66666846218815139649055310116, −6.12218458892114038725841011913, −5.40693057608805003867521193133, −4.48520069330489938764571763709, −3.06521475095947448569175146778, −1.95578321239629902702917705700, 0.34442123338273267153958013544, 2.96782362099514161978538598364, 4.02013317965774042954685030322, 5.13543628997235737582653200757, 5.61113778608919402343563673922, 7.27217759478324430867146266115, 7.54329052500876665601616025774, 8.542397851800367389900623835244, 9.995573386891773539033731680231, 10.61667032719612308804717391794

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