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Properties

Label 2-546-91.30-c1-0-4
Degree $2$
Conductor $546$
Sign $0.584 - 0.811i$
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  + (−0.866 + 0.5i)2-s + 3-s + (0.499 − 0.866i)4-s + (0.910 + 0.525i)5-s + (−0.866 + 0.5i)6-s + (2.61 + 0.397i)7-s + 0.999i·8-s + 9-s − 1.05·10-s + 6.56i·11-s + (0.499 − 0.866i)12-s + (−3.07 − 1.88i)13-s + (−2.46 + 0.963i)14-s + (0.910 + 0.525i)15-s + (−0.5 − 0.866i)16-s + (2.31 − 4.00i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + 0.577·3-s + (0.249 − 0.433i)4-s + (0.407 + 0.234i)5-s + (−0.353 + 0.204i)6-s + (0.988 + 0.150i)7-s + 0.353i·8-s + 0.333·9-s − 0.332·10-s + 1.97i·11-s + (0.144 − 0.249i)12-s + (−0.852 − 0.522i)13-s + (−0.658 + 0.257i)14-s + (0.234 + 0.135i)15-s + (−0.125 − 0.216i)16-s + (0.560 − 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.584 - 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35775 + 0.695469i\)
\(L(\frac12)\) \(\approx\) \(1.35775 + 0.695469i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 - T \)
7 \( 1 + (-2.61 - 0.397i)T \)
13 \( 1 + (3.07 + 1.88i)T \)
good5 \( 1 + (-0.910 - 0.525i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 6.56iT - 11T^{2} \)
17 \( 1 + (-2.31 + 4.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 2.99iT - 19T^{2} \)
23 \( 1 + (-1.59 - 2.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.12 - 1.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.58 + 4.38i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.02 - 2.32i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.65 - 3.84i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.696 - 1.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.01 + 5.20i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.40 - 5.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.16 + 1.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 9.90T + 61T^{2} \)
67 \( 1 + 8.98iT - 67T^{2} \)
71 \( 1 + (-7.13 + 4.11i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.18 + 4.72i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.65 + 2.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.79iT - 83T^{2} \)
89 \( 1 + (3.76 - 2.17i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.83 + 3.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.57870650510577161631142496678, −9.777571061022506584752884694265, −9.420158958509235642351286630734, −7.949137219295680212436970507648, −7.67964651704232363716313095976, −6.68422172591410872845554159168, −5.29082469238274921171395397058, −4.51789409570685082460373593798, −2.65335746797914749333121108238, −1.68354186873203268593365666401, 1.15999725058768827140993238969, 2.47627698159272502351732749899, 3.67413639612896060764581094305, 4.96859546110879431865223572964, 6.14438803192411831124394634904, 7.37901904903129417753823202539, 8.330524257001536658893403466200, 8.742459549418728805076782421661, 9.710787395713613464615711865626, 10.69688598732630995636448676201

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