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Properties

Label 2-546-273.101-c1-0-18
Degree $2$
Conductor $546$
Sign $0.255 - 0.966i$
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 + (1.51 + 0.844i)3-s + (−0.499 − 0.866i)4-s + (0.511 − 0.295i)5-s + (−1.48 + 0.887i)6-s + (1.57 + 2.12i)7-s + 0.999·8-s + (1.57 + 2.55i)9-s + 0.590i·10-s + 6.11·11-s + (−0.0248 − 1.73i)12-s + (−1.86 − 3.08i)13-s + (−2.62 + 0.304i)14-s + (1.02 − 0.0146i)15-s + (−0.5 + 0.866i)16-s + (−3.82 − 6.62i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.873 + 0.487i)3-s + (−0.249 − 0.433i)4-s + (0.228 − 0.131i)5-s + (−0.607 + 0.362i)6-s + (0.596 + 0.802i)7-s + 0.353·8-s + (0.524 + 0.851i)9-s + 0.186i·10-s + 1.84·11-s + (−0.00716 − 0.499i)12-s + (−0.518 − 0.855i)13-s + (−0.702 + 0.0814i)14-s + (0.263 − 0.00378i)15-s + (−0.125 + 0.216i)16-s + (−0.927 − 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.255 - 0.966i$
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: \((2,\ 546,\ (\ :1/2),\ 0.255 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44609 + 1.11338i\)
\(L(\frac12)\) \(\approx\) \(1.44609 + 1.11338i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.51 - 0.844i)T \)
7 \( 1 + (-1.57 - 2.12i)T \)
13 \( 1 + (1.86 + 3.08i)T \)
good5 \( 1 + (-0.511 + 0.295i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 6.11T + 11T^{2} \)
17 \( 1 + (3.82 + 6.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 3.95T + 19T^{2} \)
23 \( 1 + (-7.26 - 4.19i)T + (11.5 + 19.9i)T^{2} \)
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 \( 1 + (1.93 + 1.11i)T + (18.5 + 32.0i)T^{2} \)
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 \( 1 + (-3.88 + 2.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 0.100iT - 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
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 \( 1 + (3.56 + 2.05i)T + (44.5 + 77.0i)T^{2} \)
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

−10.91718394098080692458082815718, −9.485324059213140333569054484158, −9.282427449532903792339198151550, −8.534424164454124081747646454994, −7.50197044717401293337187855654, −6.62487149624221331223354282984, −5.27778605843825278018021738993, −4.56688561983859281352457434281, −3.11315445004864470543981300636, −1.70457420766540552689285367648, 1.35362472173947099910960236988, 2.24500120616014495103242823337, 3.90040411612903148168152055094, 4.33053854235898182145661931144, 6.60891865121393855254517121215, 6.87506864568497890474039420520, 8.319216638107427853396336713353, 8.762763168915309150481687032018, 9.651651804100490688566275975389, 10.60243107179829782118095425746

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