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Properties

Label 2-546-91.76-c1-0-12
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: $\chi_{546} (349, \cdot )$
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

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

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