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Properties

Label 2-546-91.30-c1-0-6
Degree $2$
Conductor $546$
Sign $0.949 - 0.312i$
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  + (−0.866 + 0.5i)2-s − 3-s + (0.499 − 0.866i)4-s + (2.34 + 1.35i)5-s + (0.866 − 0.5i)6-s + (2.61 − 0.407i)7-s + 0.999i·8-s + 9-s − 2.70·10-s − 5.09i·11-s + (−0.499 + 0.866i)12-s + (0.313 + 3.59i)13-s + (−2.06 + 1.65i)14-s + (−2.34 − 1.35i)15-s + (−0.5 − 0.866i)16-s + (2.86 − 4.96i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s − 0.577·3-s + (0.249 − 0.433i)4-s + (1.04 + 0.604i)5-s + (0.353 − 0.204i)6-s + (0.988 − 0.153i)7-s + 0.353i·8-s + 0.333·9-s − 0.854·10-s − 1.53i·11-s + (−0.144 + 0.249i)12-s + (0.0868 + 0.996i)13-s + (−0.550 + 0.443i)14-s + (−0.604 − 0.348i)15-s + (−0.125 − 0.216i)16-s + (0.695 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.949 - 0.312i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20336 + 0.192792i\)
\(L(\frac12)\) \(\approx\) \(1.20336 + 0.192792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + T \)
7 \( 1 + (-2.61 + 0.407i)T \)
13 \( 1 + (-0.313 - 3.59i)T \)
good5 \( 1 + (-2.34 - 1.35i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 5.09iT - 11T^{2} \)
17 \( 1 + (-2.86 + 4.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 4.38iT - 19T^{2} \)
23 \( 1 + (3.73 + 6.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.18 - 7.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.21 + 3.58i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.86 + 3.38i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.29 - 1.90i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.10 - 5.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.44 - 4.29i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.60 - 6.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.61 + 3.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 4.65T + 61T^{2} \)
67 \( 1 - 6.06iT - 67T^{2} \)
71 \( 1 + (-0.792 + 0.457i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.93 - 3.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.81 - 6.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 + (8.99 - 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.20 - 4.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.84345364846037107256368666633, −10.00689210623273646702770340839, −9.144592553569957092555341496936, −8.158122295279179645342519106328, −7.22950628679616851479721607120, −6.08418004110124347157876212642, −5.75788083267547405777720034545, −4.39556149285585910061759619677, −2.59503668622980124093250033322, −1.18961953355634180809476784921, 1.31417530672618486817241386344, 2.23597753137304716862048231817, 4.20029637054417319500210470796, 5.27047469990706169825324410769, 5.98619038286026048559188654780, 7.39994815343267305854649536767, 8.090315635393887397298624295404, 9.227765814525371974927028110221, 9.963298383156341212852836103648, 10.53145413772352802697635826599

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