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Properties

Label 2-546-39.2-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.925 - 0.378i$
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 + (−1.64 + 0.542i)3-s + (−0.866 − 0.499i)4-s + (1.01 + 1.01i)5-s + (−0.0985 − 1.72i)6-s + (−0.965 + 0.258i)7-s + (0.707 − 0.707i)8-s + (2.41 − 1.78i)9-s + (−1.24 + 0.716i)10-s + (5.93 + 1.59i)11-s + (1.69 + 0.352i)12-s + (−2.27 + 2.80i)13-s − i·14-s + (−2.21 − 1.11i)15-s + (0.500 + 0.866i)16-s + (−3.35 + 5.81i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.949 + 0.313i)3-s + (−0.433 − 0.249i)4-s + (0.452 + 0.452i)5-s + (−0.0402 − 0.705i)6-s + (−0.365 + 0.0978i)7-s + (0.249 − 0.249i)8-s + (0.803 − 0.595i)9-s + (−0.392 + 0.226i)10-s + (1.79 + 0.479i)11-s + (0.489 + 0.101i)12-s + (−0.629 + 0.776i)13-s − 0.267i·14-s + (−0.571 − 0.288i)15-s + (0.125 + 0.216i)16-s + (−0.813 + 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.925 - 0.378i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.925 - 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152963 + 0.777846i\)
\(L(\frac12)\) \(\approx\) \(0.152963 + 0.777846i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 0.542i)T \)
7 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (2.27 - 2.80i)T \)
good5 \( 1 + (-1.01 - 1.01i)T + 5iT^{2} \)
11 \( 1 + (-5.93 - 1.59i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.35 - 5.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0419 - 0.156i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.55 + 2.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.86 - 3.96i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.28 + 2.28i)T - 31iT^{2} \)
37 \( 1 + (1.39 - 5.21i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.728 - 2.71i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.39 + 0.804i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.71 - 1.71i)T - 47iT^{2} \)
53 \( 1 - 8.12iT - 53T^{2} \)
59 \( 1 + (1.78 + 6.67i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.36 - 7.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.05 - 1.35i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (9.93 - 2.66i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.46 - 2.46i)T + 73iT^{2} \)
79 \( 1 - 5.05T + 79T^{2} \)
83 \( 1 + (-3.93 - 3.93i)T + 83iT^{2} \)
89 \( 1 + (-10.2 - 2.75i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.05 + 3.93i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.09190232384617633374934187448, −10.14097793412148445998416775460, −9.509033185702129742690800371601, −8.726324004243927041196236544742, −7.18479157120665927622917764086, −6.42989331168395792518900263298, −6.12011935296792625040779376445, −4.64441384371542177303262096786, −3.90831240052969449218755567561, −1.72207463349628825122168734004, 0.56770980042005180865841599427, 1.90776257348126688150279554935, 3.57501861882556900044271117123, 4.78258760121130371495966526308, 5.70592768875924631966767422400, 6.71830465912979718671718613840, 7.62884128706286966682380958563, 9.113223535787192657066083753824, 9.492394627128102744302683755331, 10.50453257597165941155105354737

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