Properties

Label 2-546-273.38-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.666 - 0.745i$
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 + (0.658 + 1.60i)3-s + (−0.499 + 0.866i)4-s + (1.00 − 0.578i)5-s + (−1.05 + 1.37i)6-s + (0.296 + 2.62i)7-s − 0.999·8-s + (−2.13 + 2.11i)9-s + (1.00 + 0.578i)10-s + (1.85 − 3.21i)11-s + (−1.71 − 0.230i)12-s + (0.361 + 3.58i)13-s + (−2.12 + 1.57i)14-s + (1.58 + 1.22i)15-s + (−0.5 − 0.866i)16-s + (−0.0128 + 0.0221i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.380 + 0.924i)3-s + (−0.249 + 0.433i)4-s + (0.448 − 0.258i)5-s + (−0.431 + 0.559i)6-s + (0.111 + 0.993i)7-s − 0.353·8-s + (−0.710 + 0.703i)9-s + (0.316 + 0.182i)10-s + (0.559 − 0.969i)11-s + (−0.495 − 0.0664i)12-s + (0.100 + 0.994i)13-s + (−0.568 + 0.419i)14-s + (0.409 + 0.315i)15-s + (−0.125 − 0.216i)16-s + (−0.00310 + 0.00537i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.666 - 0.745i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.666 - 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.810945 + 1.81341i\)
\(L(\frac12)\) \(\approx\) \(0.810945 + 1.81341i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.658 - 1.60i)T \)
7 \( 1 + (-0.296 - 2.62i)T \)
13 \( 1 + (-0.361 - 3.58i)T \)
good5 \( 1 + (-1.00 + 0.578i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.85 + 3.21i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.0128 - 0.0221i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.122 - 0.211i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.36 - 1.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.51iT - 29T^{2} \)
31 \( 1 + (-2.24 + 3.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.29 + 4.79i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.471iT - 41T^{2} \)
43 \( 1 + 1.24T + 43T^{2} \)
47 \( 1 + (-0.203 + 0.117i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.44 - 3.72i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.80 - 1.61i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.2 + 5.91i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.24 + 3.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + (-0.784 + 1.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.01 - 5.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.64iT - 83T^{2} \)
89 \( 1 + (-8.11 + 4.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−11.37127823580717110593910039430, −9.924996091066767789091513140561, −9.156343783138950834837649632732, −8.695455006873448467708400069265, −7.69959027829538092565990846767, −6.14520197789995969292064192467, −5.69627277949044709831608536771, −4.55529920445653124811161509352, −3.60657899941217805975992294247, −2.27078858306348081264594677254, 1.06712027029137182699250967197, 2.29669621931408674924810930108, 3.46220715571455332109557445989, 4.62414506422921486582414706299, 5.98703879232188086947086479028, 6.83819794049396717073976867203, 7.70425262394725432235752949499, 8.697433274642466477438322072071, 9.909806457850190294186609419466, 10.36315110570496989723320468554

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