Properties

Label 2-546-273.242-c1-0-27
Degree $2$
Conductor $546$
Sign $0.723 - 0.689i$
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  + (0.965 + 0.258i)2-s + (1.61 + 0.637i)3-s + (0.866 + 0.499i)4-s + (2.88 + 0.772i)5-s + (1.39 + 1.03i)6-s + (−2.53 − 0.754i)7-s + (0.707 + 0.707i)8-s + (2.18 + 2.05i)9-s + (2.58 + 1.49i)10-s + (−0.475 + 0.127i)11-s + (1.07 + 1.35i)12-s + (−3.60 − 0.171i)13-s + (−2.25 − 1.38i)14-s + (4.14 + 3.08i)15-s + (0.500 + 0.866i)16-s + (−0.186 + 0.323i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.929 + 0.368i)3-s + (0.433 + 0.249i)4-s + (1.28 + 0.345i)5-s + (0.567 + 0.421i)6-s + (−0.958 − 0.285i)7-s + (0.249 + 0.249i)8-s + (0.729 + 0.684i)9-s + (0.816 + 0.471i)10-s + (−0.143 + 0.0384i)11-s + (0.310 + 0.391i)12-s + (−0.998 − 0.0476i)13-s + (−0.602 − 0.370i)14-s + (1.07 + 0.795i)15-s + (0.125 + 0.216i)16-s + (−0.0452 + 0.0784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.723 - 0.689i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.723 - 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.88395 + 1.15423i\)
\(L(\frac12)\) \(\approx\) \(2.88395 + 1.15423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-1.61 - 0.637i)T \)
7 \( 1 + (2.53 + 0.754i)T \)
13 \( 1 + (3.60 + 0.171i)T \)
good5 \( 1 + (-2.88 - 0.772i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.475 - 0.127i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.186 - 0.323i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.747 + 2.78i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.98 + 6.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.582iT - 29T^{2} \)
31 \( 1 + (-4.88 + 1.30i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.74 + 0.734i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.61 - 6.61i)T - 41iT^{2} \)
43 \( 1 - 2.66iT - 43T^{2} \)
47 \( 1 + (-3.42 + 12.7i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.82 - 1.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.22 - 1.66i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.77 - 6.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.82 + 0.757i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.908 - 0.908i)T - 71iT^{2} \)
73 \( 1 + (-0.884 - 3.30i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.68 - 9.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.25 + 9.25i)T - 83iT^{2} \)
89 \( 1 + (0.844 - 3.15i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (12.1 + 12.1i)T + 97iT^{2} \)
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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

−10.46178979287915679999910601653, −10.09283226321318920332054019155, −9.341347115279688496063839886404, −8.248729709722367820219969722584, −7.05335853269953278480317211815, −6.41660816959991167572343972112, −5.25050837831236567369679208594, −4.19775107556621463261056814086, −2.90815899092765889990905086774, −2.26204504866555889502338482041, 1.72433559894574444331960686137, 2.66576088193795951542696348130, 3.71027349727587321572163201281, 5.16948328113493566062209829834, 6.04324297226636295520589097702, 6.91470560308267103204087751269, 7.969054386372990755145356865333, 9.276943258683120084747333571007, 9.668233400630978365781539208368, 10.40447836011844938400172039941

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