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Properties

Label 2-546-91.73-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.999 - 0.00282i$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)3-s + (−0.866 − 0.499i)4-s + (−1.01 − 0.272i)5-s + (0.707 − 0.707i)6-s + (0.109 − 2.64i)7-s + (0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.526 − 0.912i)10-s + (0.304 + 1.13i)11-s + (0.5 + 0.866i)12-s + (−0.201 + 3.59i)13-s + (2.52 + 0.789i)14-s + (0.744 + 0.744i)15-s + (0.500 + 0.866i)16-s + (−2.53 + 4.39i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.499 − 0.288i)3-s + (−0.433 − 0.249i)4-s + (−0.455 − 0.121i)5-s + (0.288 − 0.288i)6-s + (0.0413 − 0.999i)7-s + (0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (0.166 − 0.288i)10-s + (0.0916 + 0.342i)11-s + (0.144 + 0.249i)12-s + (−0.0559 + 0.998i)13-s + (0.674 + 0.211i)14-s + (0.192 + 0.192i)15-s + (0.125 + 0.216i)16-s + (−0.614 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.999 - 0.00282i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.999 - 0.00282i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000240589 + 0.170272i\)
\(L(\frac12)\) \(\approx\) \(0.000240589 + 0.170272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.109 + 2.64i)T \)
13 \( 1 + (0.201 - 3.59i)T \)
good5 \( 1 + (1.01 + 0.272i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.304 - 1.13i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.53 - 4.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.78 + 1.54i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.58 - 3.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 + (1.01 + 3.79i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.35 - 0.363i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.05 - 2.05i)T - 41iT^{2} \)
43 \( 1 - 1.78iT - 43T^{2} \)
47 \( 1 + (1.77 - 6.63i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.443 - 0.768i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.61 - 1.77i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.72 - 5.61i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.4 + 3.07i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.62 - 4.62i)T + 71iT^{2} \)
73 \( 1 + (8.65 - 2.31i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.14 + 5.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.80 + 9.80i)T - 83iT^{2} \)
89 \( 1 + (1.89 - 7.08i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.59 + 1.59i)T - 97iT^{2} \)
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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

−11.15845342104501185711637061097, −10.38116258854709765283001671738, −9.452961086808722584647642671808, −8.302802517179903959051286185869, −7.63664978068865078709828778863, −6.65489532269812783147320287717, −6.09269436199108518840200396735, −4.47339259700418492333981498523, −4.11238082922850263659929109148, −1.76052702988840993055774452413, 0.10795688757413127483218940595, 2.22024861509916808244873342537, 3.41965040576155180918546309067, 4.58839074297420575762662680705, 5.58308104073238086059762347852, 6.58551885281169560568429693545, 7.986086741267046292684059278487, 8.657456444541336546398424838619, 9.625995930420002150383313303484, 10.52253200661681560109829621626

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