Properties

Label 2-546-273.194-c1-0-14
Degree $2$
Conductor $546$
Sign $0.380 + 0.924i$
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 + (−1.62 − 0.611i)3-s + (−0.499 − 0.866i)4-s + (0.0759 + 0.0438i)5-s + (1.33 − 1.09i)6-s + (−2.62 + 0.361i)7-s + 0.999·8-s + (2.25 + 1.98i)9-s + (−0.0759 + 0.0438i)10-s + (2.83 + 4.90i)11-s + (0.280 + 1.70i)12-s + (−2.43 − 2.66i)13-s + (0.997 − 2.45i)14-s + (−0.0962 − 0.117i)15-s + (−0.5 + 0.866i)16-s + (−1.33 − 2.31i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.935 − 0.352i)3-s + (−0.249 − 0.433i)4-s + (0.0339 + 0.0196i)5-s + (0.546 − 0.448i)6-s + (−0.990 + 0.136i)7-s + 0.353·8-s + (0.750 + 0.660i)9-s + (−0.0240 + 0.0138i)10-s + (0.853 + 1.47i)11-s + (0.0811 + 0.493i)12-s + (−0.674 − 0.738i)13-s + (0.266 − 0.654i)14-s + (−0.0248 − 0.0303i)15-s + (−0.125 + 0.216i)16-s + (−0.323 − 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.380 + 0.924i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.380 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421509 - 0.282404i\)
\(L(\frac12)\) \(\approx\) \(0.421509 - 0.282404i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 + 0.611i)T \)
7 \( 1 + (2.62 - 0.361i)T \)
13 \( 1 + (2.43 + 2.66i)T \)
good5 \( 1 + (-0.0759 - 0.0438i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.83 - 4.90i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.33 + 2.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.32 + 5.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.06 + 0.613i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.84iT - 29T^{2} \)
31 \( 1 + (0.441 + 0.765i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.456 - 0.263i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.99iT - 41T^{2} \)
43 \( 1 + 9.69T + 43T^{2} \)
47 \( 1 + (-7.45 - 4.30i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-12.1 + 7.00i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.11 - 1.79i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.01 - 1.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.54 - 3.77i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.65T + 71T^{2} \)
73 \( 1 + (4.11 + 7.12i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.19 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.86iT - 83T^{2} \)
89 \( 1 + (2.93 + 1.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−10.30943207925732144998546167057, −9.844205635905611025847281130867, −9.017322757385581491187098555375, −7.50659390460624978144414468584, −7.03743391973323860895208087790, −6.23994828647524832153784013093, −5.21956049019457985664454025712, −4.27716028117032979630333635388, −2.33077308303378725732570590476, −0.40713792824479849809212096080, 1.31335769659648406331033218490, 3.34452627524208002275224422409, 4.01783439576150967555958205823, 5.49601506372446644587162481213, 6.32308525107481501138265145830, 7.24128393437229669417260540458, 8.647526823165791319738999261620, 9.471954803035787195809722725432, 10.10053632390810670676501614942, 10.97738173742304358263404915478

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