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Properties

Label 2-546-273.185-c1-0-19
Degree $2$
Conductor $546$
Sign $0.726 - 0.686i$
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 + (1.62 + 0.593i)3-s + (0.499 + 0.866i)4-s + (1.76 + 3.05i)5-s + (−1.11 − 1.32i)6-s + (1.45 + 2.20i)7-s − 0.999i·8-s + (2.29 + 1.93i)9-s − 3.52i·10-s − 5.58i·11-s + (0.299 + 1.70i)12-s + (1.38 − 3.32i)13-s + (−0.158 − 2.64i)14-s + (1.05 + 6.01i)15-s + (−0.5 + 0.866i)16-s + (−1.37 − 2.38i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.939 + 0.342i)3-s + (0.249 + 0.433i)4-s + (0.788 + 1.36i)5-s + (−0.454 − 0.542i)6-s + (0.550 + 0.834i)7-s − 0.353i·8-s + (0.765 + 0.643i)9-s − 1.11i·10-s − 1.68i·11-s + (0.0864 + 0.492i)12-s + (0.385 − 0.922i)13-s + (−0.0423 − 0.705i)14-s + (0.272 + 1.55i)15-s + (−0.125 + 0.216i)16-s + (−0.333 − 0.577i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.726 - 0.686i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.726 - 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66166 + 0.660755i\)
\(L(\frac12)\) \(\approx\) \(1.66166 + 0.660755i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 0.593i)T \)
7 \( 1 + (-1.45 - 2.20i)T \)
13 \( 1 + (-1.38 + 3.32i)T \)
good5 \( 1 + (-1.76 - 3.05i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 5.58iT - 11T^{2} \)
17 \( 1 + (1.37 + 2.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 7.22iT - 19T^{2} \)
23 \( 1 + (3.91 + 2.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.473 - 0.273i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.25 + 0.722i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.35 + 2.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.63 + 6.28i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0709 + 0.122i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.96 - 3.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.33 + 5.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.71 - 6.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + (-1.01 - 0.584i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-11.2 - 6.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.47 + 7.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.92T + 83T^{2} \)
89 \( 1 + (-0.677 + 1.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.1 - 6.41i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.69203176579974966119443783148, −10.11118155280755491430922876783, −9.157343129364432330424170650355, −8.320646062339333660295763060839, −7.74044342328059987226067695807, −6.33422687276686779342226253256, −5.56145296862185084850218825644, −3.61496736275512825051492994303, −2.88927842277276324606757231082, −1.93176986448001026878169971497, 1.36561572870195508080806127304, 2.05986321679128804329456679923, 4.26302240973300795431687835593, 4.87143115002545741152992847077, 6.46195245538517347349496156909, 7.26024907891111567789328234810, 8.129737871228255850089440480137, 9.001750797558154669789059306137, 9.496224737439587959507391133939, 10.25833407125531871628565502635

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