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Properties

Label 2-546-273.251-c1-0-6
Degree $2$
Conductor $546$
Sign $-0.650 - 0.759i$
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 + (1.72 − 0.182i)3-s + (−0.499 − 0.866i)4-s + 3.28i·5-s + (−0.702 + 1.58i)6-s + (−2.38 − 1.14i)7-s + 0.999·8-s + (2.93 − 0.629i)9-s + (−2.84 − 1.64i)10-s + (−2.08 + 3.61i)11-s + (−1.01 − 1.40i)12-s + (−3.60 − 0.0395i)13-s + (2.18 − 1.49i)14-s + (0.599 + 5.65i)15-s + (−0.5 + 0.866i)16-s + (3.84 + 6.66i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.994 − 0.105i)3-s + (−0.249 − 0.433i)4-s + 1.46i·5-s + (−0.286 + 0.646i)6-s + (−0.901 − 0.431i)7-s + 0.353·8-s + (0.977 − 0.209i)9-s + (−0.899 − 0.519i)10-s + (−0.628 + 1.08i)11-s + (−0.294 − 0.404i)12-s + (−0.999 − 0.0109i)13-s + (0.583 − 0.399i)14-s + (0.154 + 1.45i)15-s + (−0.125 + 0.216i)16-s + (0.933 + 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.650 - 0.759i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.650 - 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.536995 + 1.16700i\)
\(L(\frac12)\) \(\approx\) \(0.536995 + 1.16700i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.182i)T \)
7 \( 1 + (2.38 + 1.14i)T \)
13 \( 1 + (3.60 + 0.0395i)T \)
good5 \( 1 - 3.28iT - 5T^{2} \)
11 \( 1 + (2.08 - 3.61i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.84 - 6.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.26 - 2.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.82 + 2.20i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.55 + 2.05i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.97T + 31T^{2} \)
37 \( 1 + (-5.29 - 3.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.73 - 1.00i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.64 + 8.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.60iT - 47T^{2} \)
53 \( 1 - 4.48iT - 53T^{2} \)
59 \( 1 + (-1.69 + 0.978i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.38 + 3.68i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.107 + 0.0620i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.79 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.12T + 73T^{2} \)
79 \( 1 - 2.20T + 79T^{2} \)
83 \( 1 - 5.80iT - 83T^{2} \)
89 \( 1 + (5.51 + 3.18i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.26 - 5.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.47231870020417486646188173323, −10.09132381972675379455365912580, −9.636842564911720755106676837194, −8.097445532795025912504703123404, −7.57273270643525146046343869331, −6.84297313825720811308092211937, −6.02354112546678944201317299651, −4.29727750491654724050462148183, −3.23537989958794344014529565066, −2.15230661776774794875417383607, 0.73569043040790200522639799650, 2.49900210398362601640848672394, 3.33129603120677757868043149478, 4.68212153783219730293262505658, 5.55167636549435103690021419582, 7.28005079780127207686778459130, 8.136169888222077477668388130462, 8.857957436053278184479658826164, 9.656118512179926348595643523927, 9.923594170857109398375579779676

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