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Properties

Label 2-546-91.6-c1-0-6
Degree $2$
Conductor $546$
Sign $0.989 - 0.141i$
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 + (−0.413 − 0.413i)5-s + (0.258 − 0.965i)6-s + (1.23 + 2.34i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−0.292 + 0.506i)10-s + (3.08 − 0.825i)11-s − 12-s + (−3.19 + 1.66i)13-s + (1.94 − 1.79i)14-s + (−0.151 − 0.565i)15-s + (0.500 − 0.866i)16-s + (1.52 + 2.63i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.499 + 0.288i)3-s + (−0.433 + 0.249i)4-s + (−0.185 − 0.185i)5-s + (0.105 − 0.394i)6-s + (0.465 + 0.884i)7-s + (0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.0925 + 0.160i)10-s + (0.928 − 0.248i)11-s − 0.288·12-s + (−0.886 + 0.462i)13-s + (0.519 − 0.480i)14-s + (−0.0391 − 0.145i)15-s + (0.125 − 0.216i)16-s + (0.369 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.989 - 0.141i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53032 + 0.108679i\)
\(L(\frac12)\) \(\approx\) \(1.53032 + 0.108679i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 2.34i)T \)
13 \( 1 + (3.19 - 1.66i)T \)
good5 \( 1 + (0.413 + 0.413i)T + 5iT^{2} \)
11 \( 1 + (-3.08 + 0.825i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.52 - 2.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.927 - 3.46i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-6.31 - 3.64i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.955 + 1.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.43 - 1.43i)T + 31iT^{2} \)
37 \( 1 + (-8.56 + 2.29i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.53 - 0.678i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.19 - 2.42i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.97 + 6.97i)T - 47iT^{2} \)
53 \( 1 - 1.08T + 53T^{2} \)
59 \( 1 + (-0.426 - 0.114i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (12.5 - 7.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.77 + 14.1i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.82 - 1.02i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.831 - 0.831i)T - 73iT^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 + (8.79 + 8.79i)T + 83iT^{2} \)
89 \( 1 + (0.813 + 3.03i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.09 + 11.5i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.81743197923134193270787166982, −9.831614359194109236735100327956, −9.100910355201461507405141967772, −8.446176872324323009767367488538, −7.55025423739427297410123801484, −6.14623188284559149227456213355, −4.92360725117715822344756172632, −3.99901908370729532152112064665, −2.80965556136261826549572050384, −1.59693914824895305017308041585, 1.03850753604372783246628245842, 2.88145980042888943802693857787, 4.24692710058192668878865466356, 5.10575233974039921464620826394, 6.63531053557384596466037930464, 7.21779035507861695546751410275, 7.88627421478208486738786730713, 8.967689255489635164929683013717, 9.659883730904714274400368188939, 10.68719754629777882652338342647

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