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Properties

Label 2-546-273.173-c1-0-8
Degree $2$
Conductor $546$
Sign $-0.830 - 0.556i$
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 + (1.23 + 1.21i)3-s + (−0.499 + 0.866i)4-s + (−0.567 − 0.327i)5-s + (−0.436 + 1.67i)6-s + (−2.19 + 1.46i)7-s − 0.999·8-s + (0.0419 + 2.99i)9-s − 0.655i·10-s + 3.09·11-s + (−1.66 + 0.459i)12-s + (−3.50 + 0.844i)13-s + (−2.37 − 1.17i)14-s + (−0.301 − 1.09i)15-s + (−0.5 − 0.866i)16-s + (−3.51 + 6.08i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.712 + 0.702i)3-s + (−0.249 + 0.433i)4-s + (−0.253 − 0.146i)5-s + (−0.178 + 0.684i)6-s + (−0.831 + 0.555i)7-s − 0.353·8-s + (0.0139 + 0.999i)9-s − 0.207i·10-s + 0.932·11-s + (−0.482 + 0.132i)12-s + (−0.972 + 0.234i)13-s + (−0.634 − 0.312i)14-s + (−0.0778 − 0.282i)15-s + (−0.125 − 0.216i)16-s + (−0.851 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.830 - 0.556i$
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: \((2,\ 546,\ (\ :1/2),\ -0.830 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.497182 + 1.63444i\)
\(L(\frac12)\) \(\approx\) \(0.497182 + 1.63444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.21i)T \)
7 \( 1 + (2.19 - 1.46i)T \)
13 \( 1 + (3.50 - 0.844i)T \)
good5 \( 1 + (0.567 + 0.327i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.09T + 11T^{2} \)
17 \( 1 + (3.51 - 6.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 6.51T + 19T^{2} \)
23 \( 1 + (1.24 - 0.717i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.89 - 2.24i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.26 + 3.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.56 + 1.48i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.52 - 4.34i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0380 + 0.0658i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.04 - 4.64i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.68 + 5.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.13 + 3.53i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 15.3iT - 61T^{2} \)
67 \( 1 - 3.99iT - 67T^{2} \)
71 \( 1 + (0.469 + 0.813i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.44 - 9.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.40 - 2.42i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.24iT - 83T^{2} \)
89 \( 1 + (9.46 - 5.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.57 - 14.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.18983145396686660416096781568, −9.868028940634260499618702654172, −9.381853121329332708873899150435, −8.536639235476342560436844681008, −7.65087725448342276094827240196, −6.57803163616620393196913550935, −5.60406616149590471391634747406, −4.39930741929995173706369363360, −3.66759830729104885658609953418, −2.42862617995506836888228031961, 0.827974444073954366167931619439, 2.53049785293219894010858835910, 3.37177099856725085056056331482, 4.41884935204920625230814195577, 5.87435615148194161707130636383, 7.15337272756677913669045784755, 7.36342389339308865846434801472, 9.040553388181166228640948761578, 9.436477948801744032669718734121, 10.38152260089799494976066151926

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