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Properties

Label 2-546-91.76-c1-0-6
Degree $2$
Conductor $546$
Sign $0.218 - 0.975i$
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 + (−2.23 − 1.41i)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 + (−0.845 − 0.533i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.218 - 0.975i$
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: \((2,\ 546,\ (\ :1/2),\ 0.218 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823295 + 0.659545i\)
\(L(\frac12)\) \(\approx\) \(0.823295 + 0.659545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 1.41i)T \)
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

−10.84247423149454023513073085386, −9.984714106944198581896911659205, −9.220043794985329830921080560261, −8.470012398796342365117837560657, −7.04585653668662481853335727215, −6.53316647719191131769745183513, −5.65272953122359986003205916603, −4.40774655583312552839531010453, −3.55031276235124058843767089600, −1.22456015830453437602378057949, 0.869355600029179889354866723175, 2.55152405191879902500787837590, 3.59396925139763139533821314746, 4.96811202767818576479392591858, 6.13053747291834072943719981462, 6.77967705580463103437613446439, 8.073371188502270600208218792605, 9.213196027996885432292321052628, 9.588782906487914595785077418033, 10.89191771507643960396150073908

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