Properties

Label 2-546-91.16-c1-0-11
Degree $2$
Conductor $546$
Sign $0.948 + 0.315i$
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  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (0.611 − 1.05i)5-s + (−0.5 + 0.866i)6-s + (1.15 − 2.38i)7-s + 8-s + (−0.499 − 0.866i)9-s + (0.611 − 1.05i)10-s + (−0.0702 + 0.121i)11-s + (−0.5 + 0.866i)12-s + (2.39 − 2.69i)13-s + (1.15 − 2.38i)14-s + (0.611 + 1.05i)15-s + 16-s + 0.186·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (0.273 − 0.473i)5-s + (−0.204 + 0.353i)6-s + (0.435 − 0.900i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.193 − 0.334i)10-s + (−0.0211 + 0.0367i)11-s + (−0.144 + 0.249i)12-s + (0.663 − 0.747i)13-s + (0.307 − 0.636i)14-s + (0.157 + 0.273i)15-s + 0.250·16-s + 0.0452·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.948 + 0.315i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.948 + 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19045 - 0.354868i\)
\(L(\frac12)\) \(\approx\) \(2.19045 - 0.354868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.15 + 2.38i)T \)
13 \( 1 + (-2.39 + 2.69i)T \)
good5 \( 1 + (-0.611 + 1.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.0702 - 0.121i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.186T + 17T^{2} \)
19 \( 1 + (0.447 + 0.775i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.0364T + 23T^{2} \)
29 \( 1 + (-2.99 - 5.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.82 + 3.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.363T + 37T^{2} \)
41 \( 1 + (-1.70 - 2.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.06 - 3.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.358 + 0.621i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.49 - 6.05i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.99T + 59T^{2} \)
61 \( 1 + (0.186 + 0.323i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.42 + 4.20i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.31 - 9.20i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.80 + 8.32i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.94 - 5.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.14T + 83T^{2} \)
89 \( 1 + 6.35T + 89T^{2} \)
97 \( 1 + (-3.24 + 5.61i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.82729666410424735545549111188, −10.15490811025180800512367276225, −9.040724652881459247367499099755, −8.020042383026674494087417554417, −7.00428051324115455320337470989, −5.90061984418864172318399969616, −5.04406999972493607819709474507, −4.21559989987828443064591951247, −3.13723280502099414913129093500, −1.26343247686759808623242967764, 1.76861868489595384093624899460, 2.82370342219578301511791916773, 4.26187232002925049303997346899, 5.41583665357631953454944774672, 6.20118394175731182585878176221, 6.92749361817609466602920348474, 8.101661550328299378654755827919, 8.954378705067371157316999199510, 10.22097159585024897851973614022, 11.10002477822043827621595765576

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