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Properties

Label 2-546-7.4-c1-0-9
Degree $2$
Conductor $546$
Sign $-0.198 + 0.980i$
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 + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.292 + 0.507i)5-s − 0.999·6-s + (1.62 − 2.09i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.292 − 0.507i)10-s + (0.207 − 0.358i)11-s + (0.499 + 0.866i)12-s + 13-s + (−2.62 − 0.358i)14-s + 0.585·15-s + (−0.5 − 0.866i)16-s + (1.20 − 2.09i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.130 + 0.226i)5-s − 0.408·6-s + (0.612 − 0.790i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.0926 − 0.160i)10-s + (0.0624 − 0.108i)11-s + (0.144 + 0.249i)12-s + 0.277·13-s + (−0.700 − 0.0958i)14-s + 0.151·15-s + (−0.125 − 0.216i)16-s + (0.292 − 0.507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.879040 - 1.07442i\)
\(L(\frac12)\) \(\approx\) \(0.879040 - 1.07442i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 2.09i)T \)
13 \( 1 - T \)
good5 \( 1 + (-0.292 - 0.507i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.207 + 0.358i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.20 + 2.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.08 - 1.88i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.82T + 29T^{2} \)
31 \( 1 + (-4.24 + 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.707 + 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 - 6.48T + 43T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.74 - 8.21i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.03 - 1.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.20 - 3.82i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.914 - 1.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 + (0.707 - 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.82 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 + (1.29 + 2.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.928T + 97T^{2} \)
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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

−10.53385908097201773531985822929, −9.821660577831951120051322361002, −8.743223353260170701923370684849, −7.925004841112605835894633203342, −7.21376738959451813231329187334, −6.07930762479780528773528171552, −4.65417418864662759117382505018, −3.55997991111337207290095287005, −2.31390321768202636331679930620, −0.976092173353726931486961429346, 1.67762116461371694294480969108, 3.29016691513859500218327276100, 4.74854054699141955874356324643, 5.40646758245438806626184321245, 6.48963198252313643067441513115, 7.66713824565087876907251207664, 8.533040214341072666452457228795, 9.070583009165941039957521976852, 9.981024876351206593514317371916, 10.89060273090623006307978306006

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