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Properties

Label 2-546-273.173-c1-0-16
Degree $2$
Conductor $546$
Sign $0.988 - 0.151i$
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  + (−0.5 − 0.866i)2-s + (−0.991 + 1.41i)3-s + (−0.499 + 0.866i)4-s + (3.72 + 2.14i)5-s + (1.72 + 0.148i)6-s + (−1.08 − 2.41i)7-s + 0.999·8-s + (−1.03 − 2.81i)9-s − 4.29i·10-s + 4.52·11-s + (−0.733 − 1.56i)12-s + (2.01 − 2.98i)13-s + (−1.54 + 2.14i)14-s + (−6.74 + 3.15i)15-s + (−0.5 − 0.866i)16-s + (0.0962 − 0.166i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.572 + 0.819i)3-s + (−0.249 + 0.433i)4-s + (1.66 + 0.960i)5-s + (0.704 + 0.0608i)6-s + (−0.410 − 0.911i)7-s + 0.353·8-s + (−0.344 − 0.938i)9-s − 1.35i·10-s + 1.36·11-s + (−0.211 − 0.452i)12-s + (0.559 − 0.828i)13-s + (−0.413 + 0.573i)14-s + (−1.74 + 0.814i)15-s + (−0.125 − 0.216i)16-s + (0.0233 − 0.0404i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.988 - 0.151i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.988 - 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34041 + 0.102349i\)
\(L(\frac12)\) \(\approx\) \(1.34041 + 0.102349i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.991 - 1.41i)T \)
7 \( 1 + (1.08 + 2.41i)T \)
13 \( 1 + (-2.01 + 2.98i)T \)
good5 \( 1 + (-3.72 - 2.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 4.52T + 11T^{2} \)
17 \( 1 + (-0.0962 + 0.166i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.69T + 19T^{2} \)
23 \( 1 + (1.22 - 0.704i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.66 - 5.00i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.16 + 2.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.93 - 2.85i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.24 - 0.717i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.73 - 2.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.70 + 5.02i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.59 + 3.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.17 + 1.83i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.11iT - 61T^{2} \)
67 \( 1 + 5.05iT - 67T^{2} \)
71 \( 1 + (-5.59 - 9.68i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.90 - 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.93 - 3.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.15iT - 83T^{2} \)
89 \( 1 + (-1.87 + 1.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.86 + 11.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.69588859232302258810304267647, −9.958931995978005935426315772032, −9.615383413781014181966306112451, −8.559950614131190632598218256326, −6.76298911828546292318273547514, −6.41809243694316613411960804230, −5.24566244371818433222161139626, −3.85171236289176671643931540645, −3.00548311180279371131759599730, −1.30372610645185341828594188920, 1.24765715515951508076537890783, 2.18594550196601246053582632527, 4.58369411807323427242757623282, 5.63468223950141845572932801833, 6.29572297454457031534345924035, 6.67624530260242355441552740513, 8.371840733655229241119745460584, 8.960064134549554169475207867474, 9.582130488668983995647679945053, 10.61373210324402256083002196711

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