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Properties

Label 2-546-273.68-c1-0-22
Degree $2$
Conductor $546$
Sign $0.839 - 0.543i$
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  + i·2-s + (1.63 − 0.562i)3-s − 4-s + (0.604 − 1.04i)5-s + (0.562 + 1.63i)6-s + (0.374 + 2.61i)7-s − i·8-s + (2.36 − 1.84i)9-s + (1.04 + 0.604i)10-s + (1.03 + 0.599i)11-s + (−1.63 + 0.562i)12-s + (1.92 − 3.04i)13-s + (−2.61 + 0.374i)14-s + (0.401 − 2.05i)15-s + 16-s + 5.23·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.945 − 0.324i)3-s − 0.5·4-s + (0.270 − 0.468i)5-s + (0.229 + 0.668i)6-s + (0.141 + 0.989i)7-s − 0.353i·8-s + (0.789 − 0.614i)9-s + (0.331 + 0.191i)10-s + (0.313 + 0.180i)11-s + (−0.472 + 0.162i)12-s + (0.533 − 0.845i)13-s + (−0.699 + 0.100i)14-s + (0.103 − 0.530i)15-s + 0.250·16-s + 1.27·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99439 + 0.589406i\)
\(L(\frac12)\) \(\approx\) \(1.99439 + 0.589406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.63 + 0.562i)T \)
7 \( 1 + (-0.374 - 2.61i)T \)
13 \( 1 + (-1.92 + 3.04i)T \)
good5 \( 1 + (-0.604 + 1.04i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.03 - 0.599i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 + (6.37 - 3.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.62iT - 23T^{2} \)
29 \( 1 + (1.43 - 0.827i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.45 - 1.99i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.00T + 37T^{2} \)
41 \( 1 + (0.0914 + 0.158i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.527 + 0.913i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.56 + 6.17i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.91 - 1.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + (5.72 - 3.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.37 - 9.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.08 + 3.51i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.40 + 3.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.23 + 3.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 + 7.19T + 89T^{2} \)
97 \( 1 + (11.8 + 6.86i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.63882322424138487496181374101, −9.649144043497093590266978054488, −8.833221723031771971345082676191, −8.313147704748503332239742152094, −7.49520863271803823507544810114, −6.24906336443720062950009978270, −5.54320059085287562744287354069, −4.23753461036907030919652602240, −3.00366716476971068169224264903, −1.52866984035952793231524767038, 1.50035087154302123066187810610, 2.80497446156178412638945762418, 3.87113099907566399093352763155, 4.55011056435715817491226492013, 6.19897503009632244655330772942, 7.31890334395586404628972366059, 8.187596762979626746693135383877, 9.209764756274019074830002322201, 9.817706600447581516220790319093, 10.77680513252227173695838597726

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