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Properties

Label 2-546-91.41-c1-0-19
Degree $2$
Conductor $546$
Sign $-0.892 + 0.451i$
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  + (−0.965 − 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.499i)4-s + (−1.77 − 1.77i)5-s + (−0.965 + 0.258i)6-s + (0.546 − 2.58i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (1.25 + 2.17i)10-s + (−0.239 + 0.893i)11-s + 12-s + (−1.53 − 3.26i)13-s + (−1.19 + 2.35i)14-s + (−2.42 − 0.649i)15-s + (0.500 + 0.866i)16-s + (−2.62 + 4.54i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 + 0.249i)4-s + (−0.793 − 0.793i)5-s + (−0.394 + 0.105i)6-s + (0.206 − 0.978i)7-s + (−0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (0.396 + 0.687i)10-s + (−0.0721 + 0.269i)11-s + 0.288·12-s + (−0.426 − 0.904i)13-s + (−0.320 + 0.630i)14-s + (−0.625 − 0.167i)15-s + (0.125 + 0.216i)16-s + (−0.636 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.892 + 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.173807 - 0.727579i\)
\(L(\frac12)\) \(\approx\) \(0.173807 - 0.727579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.546 + 2.58i)T \)
13 \( 1 + (1.53 + 3.26i)T \)
good5 \( 1 + (1.77 + 1.77i)T + 5iT^{2} \)
11 \( 1 + (0.239 - 0.893i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.62 - 4.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.76 + 0.472i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.27 - 3.04i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.15 + 5.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.75 + 3.75i)T + 31iT^{2} \)
37 \( 1 + (0.542 - 2.02i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.84 + 6.90i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.61 - 4.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.17 + 2.17i)T - 47iT^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 + (1.46 + 5.48i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.63 + 2.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.93 - 1.32i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.60 + 13.4i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.41 - 4.41i)T - 73iT^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + (-1.26 - 1.26i)T + 83iT^{2} \)
89 \( 1 + (-7.69 - 2.06i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.07 + 0.556i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.33009145512067163887506624869, −9.515781900914577373718232568185, −8.492695020346307148895506600365, −7.77730615998491954298497885935, −7.38154248998574391119526400792, −5.93393145629929223841769715230, −4.41207511229300754251081135050, −3.62996377855673721766778704218, −1.98038328779537641284313363536, −0.49006797190522619121550932925, 2.17084974282724989770157442205, 3.16690842334718656764359163502, 4.51047925635443951926882094079, 5.78412302496851988506305161790, 7.02931278402619045096806868917, 7.57075920527899298510071693041, 8.722614948627143317010542460573, 9.166735881001966001574804395809, 10.20202484512380756163654076399, 11.20614168749185016716279406307

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