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Properties

Label 2-546-91.73-c1-0-18
Degree $2$
Conductor $546$
Sign $0.310 + 0.950i$
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 + (−1.56 − 0.419i)5-s + (−0.707 + 0.707i)6-s + (−2.13 − 1.56i)7-s + (0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.810 − 1.40i)10-s + (−0.354 − 1.32i)11-s + (−0.5 − 0.866i)12-s + (−2.72 − 2.36i)13-s + (2.06 − 1.65i)14-s + (−1.14 − 1.14i)15-s + (0.500 + 0.866i)16-s + (1.90 − 3.29i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (−0.700 − 0.187i)5-s + (−0.288 + 0.288i)6-s + (−0.805 − 0.591i)7-s + (0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (0.256 − 0.444i)10-s + (−0.106 − 0.398i)11-s + (−0.144 − 0.249i)12-s + (−0.754 − 0.656i)13-s + (0.551 − 0.442i)14-s + (−0.296 − 0.296i)15-s + (0.125 + 0.216i)16-s + (0.460 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.310 + 0.950i$
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: \((2,\ 546,\ (\ :1/2),\ 0.310 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557771 - 0.404483i\)
\(L(\frac12)\) \(\approx\) \(0.557771 - 0.404483i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 + 1.56i)T \)
13 \( 1 + (2.72 + 2.36i)T \)
good5 \( 1 + (1.56 + 0.419i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.354 + 1.32i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.90 + 3.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.67 + 0.448i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.60 + 2.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.0101T + 29T^{2} \)
31 \( 1 + (2.23 + 8.33i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (5.33 + 1.42i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.53 + 1.53i)T - 41iT^{2} \)
43 \( 1 - 7.52iT - 43T^{2} \)
47 \( 1 + (0.746 - 2.78i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.77 + 3.07i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.53 - 1.48i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.00 - 0.579i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.07 - 0.555i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.80 - 7.80i)T + 71iT^{2} \)
73 \( 1 + (1.16 - 0.311i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.13 + 12.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.61 + 3.61i)T - 83iT^{2} \)
89 \( 1 + (-1.84 + 6.89i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.73 - 5.73i)T - 97iT^{2} \)
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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

−10.34724648404558469474826143090, −9.647811656077844517377442746235, −8.805652953008920834413059901907, −7.77318666550089233120921963395, −7.32117792934749031876597329916, −6.13203008532524311061904158860, −4.94004420163598208641769065731, −3.94646460382520233598374597143, −2.87712995670840693169551507630, −0.38909891703415361573438435828, 1.86256941603511528052581638443, 3.08107660938452444029986117829, 3.91290002421427834847218888030, 5.25253239075741111678118458988, 6.68771734372632746260924274917, 7.47078198854290268338285222841, 8.502254697716235080845040408125, 9.237700226979371476794804647137, 10.04644644446725190213886618363, 10.95569662884430243970570644335

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