Properties

Label 2-546-273.101-c1-0-25
Degree $2$
Conductor $546$
Sign $0.477 + 0.878i$
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 + (0.428 − 1.67i)3-s + (−0.499 − 0.866i)4-s + (1.58 − 0.917i)5-s + (1.23 + 1.20i)6-s + (2.08 − 1.62i)7-s + 0.999·8-s + (−2.63 − 1.43i)9-s + 1.83i·10-s + 3.39·11-s + (−1.66 + 0.468i)12-s + (−3.07 − 1.88i)13-s + (0.364 + 2.62i)14-s + (−0.860 − 3.06i)15-s + (−0.5 + 0.866i)16-s + (1.59 + 2.75i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.247 − 0.968i)3-s + (−0.249 − 0.433i)4-s + (0.711 − 0.410i)5-s + (0.506 + 0.493i)6-s + (0.788 − 0.614i)7-s + 0.353·8-s + (−0.877 − 0.478i)9-s + 0.580i·10-s + 1.02·11-s + (−0.481 + 0.135i)12-s + (−0.851 − 0.523i)13-s + (0.0973 + 0.700i)14-s + (−0.222 − 0.790i)15-s + (−0.125 + 0.216i)16-s + (0.386 + 0.668i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28491 - 0.763756i\)
\(L(\frac12)\) \(\approx\) \(1.28491 - 0.763756i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.428 + 1.67i)T \)
7 \( 1 + (-2.08 + 1.62i)T \)
13 \( 1 + (3.07 + 1.88i)T \)
good5 \( 1 + (-1.58 + 0.917i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 3.39T + 11T^{2} \)
17 \( 1 + (-1.59 - 2.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.03T + 19T^{2} \)
23 \( 1 + (2.14 + 1.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.81 + 1.04i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.14 + 5.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.436 + 0.251i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.35 - 2.51i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.528 + 0.916i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.0 + 5.82i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.81 - 3.35i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.37 - 3.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 11.6iT - 61T^{2} \)
67 \( 1 - 9.92iT - 67T^{2} \)
71 \( 1 + (7.77 - 13.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.47 + 7.75i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.92 - 6.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.0iT - 83T^{2} \)
89 \( 1 + (-4.61 - 2.66i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.79 + 8.31i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.45810869184007301821710493409, −9.600996905731823948537967374443, −8.644720401631244850867349337204, −7.955764117040062577448648325327, −7.12995928362870938903110002236, −6.21129103318854397215773164055, −5.35564995673985068149883601519, −4.05271116183159371051070961872, −2.14362656870750145778829625513, −1.02924785669726818665720386797, 1.91854672557863405117977323879, 2.90389369387509782121442189487, 4.24077174443001288017564412526, 5.11439463002796078766676440594, 6.27945344661417623751503899348, 7.60429571512561431161168055790, 8.759334327225192986155058276225, 9.247269534720834543947761428945, 10.08990473960064466261740882205, 10.72955444461193647526390222762

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