Properties

Label 2-546-273.38-c1-0-34
Degree $2$
Conductor $546$
Sign $-0.0485 + 0.998i$
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 + (0.806 − 1.53i)3-s + (−0.499 + 0.866i)4-s + (−3.23 + 1.86i)5-s + (1.73 − 0.0678i)6-s + (−0.481 − 2.60i)7-s − 0.999·8-s + (−1.69 − 2.47i)9-s + (−3.23 − 1.86i)10-s + (0.664 − 1.15i)11-s + (0.924 + 1.46i)12-s + (−0.452 − 3.57i)13-s + (2.01 − 1.71i)14-s + (0.253 + 6.46i)15-s + (−0.5 − 0.866i)16-s + (2.49 − 4.31i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.465 − 0.884i)3-s + (−0.249 + 0.433i)4-s + (−1.44 + 0.835i)5-s + (0.706 − 0.0276i)6-s + (−0.182 − 0.983i)7-s − 0.353·8-s + (−0.566 − 0.824i)9-s + (−1.02 − 0.590i)10-s + (0.200 − 0.346i)11-s + (0.266 + 0.422i)12-s + (−0.125 − 0.992i)13-s + (0.537 − 0.459i)14-s + (0.0653 + 1.66i)15-s + (−0.125 − 0.216i)16-s + (0.604 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0485 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0485 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0485 + 0.998i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0485 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693096 - 0.727637i\)
\(L(\frac12)\) \(\approx\) \(0.693096 - 0.727637i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.806 + 1.53i)T \)
7 \( 1 + (0.481 + 2.60i)T \)
13 \( 1 + (0.452 + 3.57i)T \)
good5 \( 1 + (3.23 - 1.86i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.664 + 1.15i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.49 + 4.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.36 + 4.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.00 - 0.578i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.29iT - 29T^{2} \)
31 \( 1 + (1.66 - 2.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.87 + 4.54i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.38iT - 41T^{2} \)
43 \( 1 + 9.54T + 43T^{2} \)
47 \( 1 + (6.95 - 4.01i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.36 + 0.788i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.31 + 5.37i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.85 + 2.80i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.51 - 2.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.95T + 71T^{2} \)
73 \( 1 + (-7.21 + 12.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.22 - 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.74iT - 83T^{2} \)
89 \( 1 + (-2.55 + 1.47i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.79T + 97T^{2} \)
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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

−10.88092837241006965457094246157, −9.531154001982796126950641016266, −8.252503262409708790318957838646, −7.72933673610193184756296164582, −7.05730871216645403731304229047, −6.44226837380187251297284203987, −4.86667318320870562315181044988, −3.52365504374624407004710628583, −3.05756786890781010366210327280, −0.48582479232466570135450847648, 2.03704538106290216861837290185, 3.56426237278942441493021926721, 4.16499419981107540910988089185, 5.00928722292844866278605104055, 6.18897171042080288072444265374, 7.963239947654721873516005946652, 8.425215461602923729899822204729, 9.347837854218893122140028770583, 10.05613755984910687970527599362, 11.26432060786846715453564798809

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