Properties

Label 2-546-13.3-c1-0-14
Degree $2$
Conductor $546$
Sign $-0.934 + 0.354i$
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 + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 2·5-s + (0.499 − 0.866i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)10-s + (−3.08 − 5.35i)11-s − 0.999·12-s + (−1.5 + 3.27i)13-s − 0.999·14-s + (−1 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.894·5-s + (0.204 − 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (−0.931 − 1.61i)11-s − 0.288·12-s + (−0.416 + 0.909i)13-s − 0.267·14-s + (−0.258 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.934 + 0.354i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.934 + 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0713627 - 0.389140i\)
\(L(\frac12)\) \(\approx\) \(0.0713627 - 0.389140i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (1.5 - 3.27i)T \)
good5 \( 1 + 2T + 5T^{2} \)
11 \( 1 + (3.08 + 5.35i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.08 + 7.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.58 + 2.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.08 + 7.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.17T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.08 + 3.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.410 - 0.711i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.17T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-0.410 + 0.711i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.58 - 13.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.58 - 7.94i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 1.82T + 79T^{2} \)
83 \( 1 + 5.17T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.17 - 10.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.63512173247268900056372132772, −9.503141534336360059508005702329, −8.675988821449877719630208959943, −8.011129274506011085100366665662, −7.09215554927866160353667894650, −5.56568902393654669087316917983, −4.33411656556602806139700567678, −3.59032881975647066795230539223, −2.39011307957279222375653869731, −0.23682064380232302450242243318, 1.90439736054073195820049045610, 3.41126679667870522850116032338, 4.85979907054153562112379120118, 5.63110288819011479644914281874, 7.23548071869529114342173401628, 7.49775028936306690129423511437, 8.190490692306051293303587545362, 9.382423608101691392648985535336, 10.06695633078835265168109457626, 11.18983502189237155005028573280

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