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Properties

Label 2-546-273.269-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.994 - 0.108i$
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 + (−1.70 − 0.301i)3-s − 4-s + (−2.12 − 3.68i)5-s + (0.301 − 1.70i)6-s + (2.63 + 0.281i)7-s − i·8-s + (2.81 + 1.02i)9-s + (3.68 − 2.12i)10-s + (−1.28 + 0.743i)11-s + (1.70 + 0.301i)12-s + (−2.41 + 2.67i)13-s + (−0.281 + 2.63i)14-s + (2.52 + 6.93i)15-s + 16-s − 2.66·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.984 − 0.173i)3-s − 0.5·4-s + (−0.952 − 1.64i)5-s + (0.122 − 0.696i)6-s + (0.994 + 0.106i)7-s − 0.353i·8-s + (0.939 + 0.342i)9-s + (1.16 − 0.673i)10-s + (−0.388 + 0.224i)11-s + (0.492 + 0.0869i)12-s + (−0.669 + 0.742i)13-s + (−0.0752 + 0.703i)14-s + (0.651 + 1.79i)15-s + 0.250·16-s − 0.646·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.994 - 0.108i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.994 - 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00367570 + 0.0672670i\)
\(L(\frac12)\) \(\approx\) \(0.00367570 + 0.0672670i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.301i)T \)
7 \( 1 + (-2.63 - 0.281i)T \)
13 \( 1 + (2.41 - 2.67i)T \)
good5 \( 1 + (2.12 + 3.68i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.28 - 0.743i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.66T + 17T^{2} \)
19 \( 1 + (3.55 + 2.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.158iT - 23T^{2} \)
29 \( 1 + (-0.413 - 0.238i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.04 - 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + (5.13 - 8.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.00 - 6.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.27 + 3.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.23 - 1.86i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.78T + 59T^{2} \)
61 \( 1 + (-2.28 - 1.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.63 - 2.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.70 - 4.44i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.03 + 3.48i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.20 + 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.37T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (3.13 - 1.81i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.57762187984981301765100468254, −10.41248022979708736947051270834, −9.146171470953789502495500600871, −8.435750309404559258225512222845, −7.65650041958489780871199884067, −6.78554707981795503077506269850, −5.45310286697749752201843159755, −4.63193105365462408377649309495, −4.43301277911700767447427433160, −1.54953523131751318803905116142, 0.04493436643734874994481628226, 2.25690233893529964628827720799, 3.58359980205723035313995384526, 4.49742462303569140212125044690, 5.60600052672007875261694095345, 6.83198795621306996922976731304, 7.57509939049357640733388731062, 8.522143253653348786047031386456, 10.23243822537867514018052715583, 10.51852214685175553485350329944

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