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Properties

Label 2-546-13.3-c1-0-15
Degree $2$
Conductor $546$
Sign $-0.522 - 0.852i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)11-s + 0.999·12-s + (−3.5 − 0.866i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + 0.999·18-s + (−2.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.204 + 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.452 − 0.783i)11-s + 0.288·12-s + (−0.970 − 0.240i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + 0.235·18-s + (−0.573 + 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (3.5 + 0.866i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.37202673827620200804606497013, −9.453642667954677657666806461203, −8.317257046262649002426373369172, −7.83873337026315190808385447530, −6.50536205682837573880549311290, −5.66492534122420757550052588777, −4.37589542230758019747744243792, −3.00578576155251164584655802355, −1.87798864913775327366154204737, 0, 2.30054975701406865933678141656, 4.05536305569503541777473301077, 4.92195010975224217259434907813, 5.87080297661846121160247021845, 7.09243760743157993969694933263, 7.55815905293563325856043959559, 8.897255014354849941293596911901, 9.651121492418243418918015037962, 10.23495964714052628902466770313

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