Properties

Label 2-546-273.257-c1-0-36
Degree $2$
Conductor $546$
Sign $-0.899 + 0.436i$
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  + 2-s + (0.243 − 1.71i)3-s + 4-s + (−2.60 − 1.50i)5-s + (0.243 − 1.71i)6-s + (−2.60 + 0.446i)7-s + 8-s + (−2.88 − 0.833i)9-s + (−2.60 − 1.50i)10-s + (2.01 − 3.49i)11-s + (0.243 − 1.71i)12-s + (0.138 + 3.60i)13-s + (−2.60 + 0.446i)14-s + (−3.21 + 4.09i)15-s + 16-s − 3.34·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.140 − 0.990i)3-s + 0.5·4-s + (−1.16 − 0.672i)5-s + (0.0992 − 0.700i)6-s + (−0.985 + 0.168i)7-s + 0.353·8-s + (−0.960 − 0.277i)9-s + (−0.823 − 0.475i)10-s + (0.607 − 1.05i)11-s + (0.0701 − 0.495i)12-s + (0.0382 + 0.999i)13-s + (−0.696 + 0.119i)14-s + (−0.828 + 1.05i)15-s + 0.250·16-s − 0.810·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.899 + 0.436i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.899 + 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.274495 - 1.19608i\)
\(L(\frac12)\) \(\approx\) \(0.274495 - 1.19608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.243 + 1.71i)T \)
7 \( 1 + (2.60 - 0.446i)T \)
13 \( 1 + (-0.138 - 3.60i)T \)
good5 \( 1 + (2.60 + 1.50i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.01 + 3.49i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.34T + 17T^{2} \)
19 \( 1 + (2.56 + 4.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.48iT - 23T^{2} \)
29 \( 1 + (-3.74 + 2.16i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.95 + 5.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.57iT - 37T^{2} \)
41 \( 1 + (-6.29 + 3.63i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.23 - 3.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.77 - 3.33i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.50 + 4.90i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.18iT - 59T^{2} \)
61 \( 1 + (5.77 - 3.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.04 - 4.07i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.436 + 0.756i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.78 + 11.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.50 + 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 + 3.63iT - 89T^{2} \)
97 \( 1 + (-1.10 + 1.91i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.88852301602414561912481207002, −9.090542722667254618396420821763, −8.698642599697686143478374904803, −7.57304683405743307104373499207, −6.61650158452540165505535497762, −6.09062112630877300533940733458, −4.54244343485144679370189415317, −3.66071778703495609529526691688, −2.44560358358059147470552682389, −0.53447847790931191079178504036, 2.71290603493578128656284059521, 3.72854369227220114195361527820, 4.16684399665426166267555294605, 5.48518478571649906957303044340, 6.63330040084636475800569989186, 7.42356436067693190868373602678, 8.511271215898637482185835305897, 9.649318099329944144587200956934, 10.48645223131806839040505021439, 11.02726569526331170430248304805

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