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Properties

Label 2-546-273.17-c1-0-12
Degree $2$
Conductor $546$
Sign $0.999 - 0.0416i$
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  + 2-s + (−1.66 − 0.459i)3-s + 4-s + (0.567 − 0.327i)5-s + (−1.66 − 0.459i)6-s + (−2.37 + 1.17i)7-s + 8-s + (2.57 + 1.53i)9-s + (0.567 − 0.327i)10-s + (1.54 + 2.67i)11-s + (−1.66 − 0.459i)12-s + (3.50 − 0.844i)13-s + (−2.37 + 1.17i)14-s + (−1.09 + 0.286i)15-s + 16-s + 7.02·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.964 − 0.265i)3-s + 0.5·4-s + (0.253 − 0.146i)5-s + (−0.681 − 0.187i)6-s + (−0.896 + 0.442i)7-s + 0.353·8-s + (0.858 + 0.512i)9-s + (0.179 − 0.103i)10-s + (0.466 + 0.807i)11-s + (−0.482 − 0.132i)12-s + (0.972 − 0.234i)13-s + (−0.634 + 0.312i)14-s + (−0.283 + 0.0738i)15-s + 0.250·16-s + 1.70·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.999 - 0.0416i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.999 - 0.0416i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74171 + 0.0362795i\)
\(L(\frac12)\) \(\approx\) \(1.74171 + 0.0362795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.66 + 0.459i)T \)
7 \( 1 + (2.37 - 1.17i)T \)
13 \( 1 + (-3.50 + 0.844i)T \)
good5 \( 1 + (-0.567 + 0.327i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.54 - 2.67i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 7.02T + 17T^{2} \)
19 \( 1 + (-3.25 + 5.64i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.43iT - 23T^{2} \)
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 \( 1 - 2.96iT - 37T^{2} \)
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 \( 1 - 7.07iT - 59T^{2} \)
61 \( 1 + (13.2 + 7.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 1.99i)T + (33.5 - 58.0i)T^{2} \)
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 \( 1 + 10.9iT - 89T^{2} \)
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

−11.11686432396216884887318152373, −9.904326819609983080318985876628, −9.416969698547092357377028433081, −7.76877479407636329807858152470, −6.94619621253580287039926000110, −5.95029035280117928039759220658, −5.49361089238560663808354529225, −4.25485154211126771845127824706, −3.01543989489878090154286756241, −1.32890729275950350782142315334, 1.18097078425202904478797824715, 3.40473041359187511000137611759, 3.91911232769111729172311117249, 5.49144693653420540723420905563, 5.99733757143179811943994409664, 6.77041999441198461416470009873, 7.889302697045944027433277158487, 9.335815137221729745941511945351, 10.21817150744750762804151680755, 10.75278694887515701375338827974

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