Properties

Label 2-546-273.251-c1-0-31
Degree $2$
Conductor $546$
Sign $-0.894 - 0.447i$
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 + (−1.63 + 0.559i)3-s + (−0.499 − 0.866i)4-s − 2.94i·5-s + (−0.334 + 1.69i)6-s + (−1.08 − 2.41i)7-s − 0.999·8-s + (2.37 − 1.83i)9-s + (−2.55 − 1.47i)10-s + (−1.48 + 2.57i)11-s + (1.30 + 1.13i)12-s + (−2.33 + 2.75i)13-s + (−2.63 − 0.272i)14-s + (1.65 + 4.83i)15-s + (−0.5 + 0.866i)16-s + (2.27 + 3.93i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.946 + 0.323i)3-s + (−0.249 − 0.433i)4-s − 1.31i·5-s + (−0.136 + 0.693i)6-s + (−0.408 − 0.912i)7-s − 0.353·8-s + (0.791 − 0.611i)9-s + (−0.807 − 0.466i)10-s + (−0.448 + 0.776i)11-s + (0.376 + 0.329i)12-s + (−0.646 + 0.762i)13-s + (−0.703 − 0.0727i)14-s + (0.426 + 1.24i)15-s + (−0.125 + 0.216i)16-s + (0.551 + 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115458 + 0.488315i\)
\(L(\frac12)\) \(\approx\) \(0.115458 + 0.488315i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 0.559i)T \)
7 \( 1 + (1.08 + 2.41i)T \)
13 \( 1 + (2.33 - 2.75i)T \)
good5 \( 1 + 2.94iT - 5T^{2} \)
11 \( 1 + (1.48 - 2.57i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.27 - 3.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.30 + 5.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.82 + 1.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.262 - 0.151i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.31T + 31T^{2} \)
37 \( 1 + (5.19 + 2.99i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.22 - 2.44i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.53 + 7.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.77iT - 47T^{2} \)
53 \( 1 + 1.95iT - 53T^{2} \)
59 \( 1 + (-5.73 + 3.30i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.23 + 3.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.91 + 3.99i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.69 + 8.13i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.10T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 5.68iT - 83T^{2} \)
89 \( 1 + (12.8 + 7.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.31 + 9.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.39753987999950149135726756920, −9.647832413306494150932588161055, −8.884749436536963272342497132061, −7.44106634311258623749942983390, −6.46955986233986287236960918199, −5.23482236243533835596362215269, −4.59718324988433970690415740190, −3.85672155095048566988998397493, −1.78210082947880392620880745796, −0.28013433777394687303762474240, 2.50203611462574290236021549492, 3.57358533972191618797302895585, 5.30647054733941663731310557504, 5.77450429057403242725330781666, 6.65848120756874672569310401062, 7.44911788491408025441063092322, 8.316525863647303812741840710930, 9.795732726750978575041361001826, 10.47329260673240254348983435922, 11.39693061320546719688503481455

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