Properties

Label 2-546-273.257-c1-0-29
Degree $2$
Conductor $546$
Sign $0.976 + 0.217i$
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 + (1.48 − 0.887i)3-s + 4-s + (0.511 + 0.295i)5-s + (1.48 − 0.887i)6-s + (2.62 + 0.304i)7-s + 8-s + (1.42 − 2.64i)9-s + (0.511 + 0.295i)10-s + (−3.05 + 5.29i)11-s + (1.48 − 0.887i)12-s + (1.86 + 3.08i)13-s + (2.62 + 0.304i)14-s + (1.02 − 0.0146i)15-s + 16-s − 7.64·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.858 − 0.512i)3-s + 0.5·4-s + (0.228 + 0.131i)5-s + (0.607 − 0.362i)6-s + (0.993 + 0.115i)7-s + 0.353·8-s + (0.474 − 0.880i)9-s + (0.161 + 0.0933i)10-s + (−0.921 + 1.59i)11-s + (0.429 − 0.256i)12-s + (0.518 + 0.855i)13-s + (0.702 + 0.0814i)14-s + (0.263 − 0.00378i)15-s + 0.250·16-s − 1.85·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.976 + 0.217i$
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: \((2,\ 546,\ (\ :1/2),\ 0.976 + 0.217i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.04372 - 0.335332i\)
\(L(\frac12)\) \(\approx\) \(3.04372 - 0.335332i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 0.887i)T \)
7 \( 1 + (-2.62 - 0.304i)T \)
13 \( 1 + (-1.86 - 3.08i)T \)
good5 \( 1 + (-0.511 - 0.295i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.05 - 5.29i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 7.64T + 17T^{2} \)
19 \( 1 + (1.97 + 3.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.38iT - 23T^{2} \)
29 \( 1 + (-1.39 + 0.803i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.38 - 2.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.22iT - 37T^{2} \)
41 \( 1 + (1.36 - 0.787i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.90 - 5.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.94 + 2.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.30 + 1.90i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.48iT - 59T^{2} \)
61 \( 1 + (-0.0871 + 0.0502i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.95 - 5.17i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.875 - 1.51i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.41 + 9.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.17 - 5.50i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.07iT - 83T^{2} \)
89 \( 1 + 4.11iT - 89T^{2} \)
97 \( 1 + (2.82 - 4.89i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.87894061597859156164649179940, −9.946215142971221559024262365563, −8.748017752245325578183299229779, −8.139248939973953517791329633744, −6.92064178610237882247402754368, −6.52430059057219440542836639017, −4.70436182455885642628857521687, −4.37164756892152480057530349498, −2.45113121666933395101025526692, −2.02320132542381051034239099631, 1.85653982554389920911378911486, 3.12108577975253776890026319971, 4.04308683969241630344104618355, 5.19887178148913703042781610353, 5.86713575207639854503634713387, 7.44097720301113382443820273901, 8.251624777926774769330648172885, 8.813570500858441250730885306230, 10.13544411589364566614606699729, 11.01294434524675559751928261341

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