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Properties

Label 2-546-91.5-c1-0-7
Degree $2$
Conductor $546$
Sign $0.508 + 0.860i$
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 + (−2.23 + 0.599i)5-s + (−0.707 − 0.707i)6-s + (1.69 + 2.03i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.15 + 2.00i)10-s + (0.875 − 3.26i)11-s + (−0.5 + 0.866i)12-s + (3.59 + 0.240i)13-s + (1.52 − 2.16i)14-s + (−1.63 + 1.63i)15-s + (0.500 − 0.866i)16-s + (−1.46 − 2.53i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.499 − 0.288i)3-s + (−0.433 + 0.249i)4-s + (−1.00 + 0.268i)5-s + (−0.288 − 0.288i)6-s + (0.640 + 0.768i)7-s + (0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.366 + 0.634i)10-s + (0.263 − 0.984i)11-s + (−0.144 + 0.249i)12-s + (0.997 + 0.0666i)13-s + (0.407 − 0.577i)14-s + (−0.423 + 0.423i)15-s + (0.125 − 0.216i)16-s + (−0.354 − 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.508 + 0.860i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.508 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24776 - 0.712090i\)
\(L(\frac12)\) \(\approx\) \(1.24776 - 0.712090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 2.03i)T \)
13 \( 1 + (-3.59 - 0.240i)T \)
good5 \( 1 + (2.23 - 0.599i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.875 + 3.26i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.46 + 2.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.28 + 1.68i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.91 - 2.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.65T + 29T^{2} \)
31 \( 1 + (-2.57 + 9.59i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.30 - 1.15i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.02 + 1.02i)T + 41iT^{2} \)
43 \( 1 - 9.35iT - 43T^{2} \)
47 \( 1 + (-1.05 - 3.94i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.66 + 2.88i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.04 + 2.42i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.98 - 2.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.43 - 2.26i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.859 + 0.859i)T - 71iT^{2} \)
73 \( 1 + (12.3 + 3.30i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.0421 - 0.0730i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.05 + 3.05i)T + 83iT^{2} \)
89 \( 1 + (1.74 + 6.52i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.13 - 1.13i)T + 97iT^{2} \)
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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

−11.10247438907323882193461366689, −9.616533332698381893618426970107, −8.830171359953789838581902848406, −8.157987635134271330144207830673, −7.39424704981364674612856569515, −6.06334022714841301374713518505, −4.76898659540933556313739459395, −3.54077660158303675182095978855, −2.78340858262141004739749993470, −1.10996353940473897735445291644, 1.30684280501510864654328852041, 3.49784610051964175807937853192, 4.33501618089777016828425854084, 5.15049711508624547750123715716, 6.75292982238524244671132698858, 7.43534402856941237342892274815, 8.313666146691987571432945635901, 8.811970319068378977058280925695, 10.06543021952829686420876029144, 10.73020910348336957043925530568

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