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Properties

Label 2-546-91.47-c1-0-14
Degree $2$
Conductor $546$
Sign $-0.331 + 0.943i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 − 0.499i)4-s + (−0.679 − 2.53i)5-s + (−0.707 + 0.707i)6-s + (−0.822 − 2.51i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.31 + 2.27i)10-s + (4.17 + 1.11i)11-s + (0.5 − 0.866i)12-s + (0.104 + 3.60i)13-s + (1.44 + 2.21i)14-s + (−1.85 − 1.85i)15-s + (0.500 − 0.866i)16-s + (−3.38 − 5.86i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 − 0.249i)4-s + (−0.304 − 1.13i)5-s + (−0.288 + 0.288i)6-s + (−0.310 − 0.950i)7-s + (−0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.415 + 0.719i)10-s + (1.25 + 0.337i)11-s + (0.144 − 0.249i)12-s + (0.0289 + 0.999i)13-s + (0.386 + 0.592i)14-s + (−0.479 − 0.479i)15-s + (0.125 − 0.216i)16-s + (−0.821 − 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.331 + 0.943i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591206 - 0.834140i\)
\(L(\frac12)\) \(\approx\) \(0.591206 - 0.834140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.822 + 2.51i)T \)
13 \( 1 + (-0.104 - 3.60i)T \)
good5 \( 1 + (0.679 + 2.53i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-4.17 - 1.11i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.38 + 5.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0834 - 0.311i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (7.77 + 4.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.03T + 29T^{2} \)
31 \( 1 + (-6.30 - 1.69i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.260 - 0.972i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-7.72 + 7.72i)T - 41iT^{2} \)
43 \( 1 + 4.30iT - 43T^{2} \)
47 \( 1 + (6.33 - 1.69i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.53 - 6.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.68 + 6.28i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.40 - 0.811i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.04 - 3.91i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.33 - 3.33i)T + 71iT^{2} \)
73 \( 1 + (0.170 - 0.636i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.04 + 5.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.56 + 6.56i)T - 83iT^{2} \)
89 \( 1 + (-6.65 + 1.78i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-10.6 + 10.6i)T - 97iT^{2} \)
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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

−10.26641435191590080283065234245, −9.253500280441199730314484352742, −9.007174387777299215026109921483, −7.906761555691116567651084034657, −7.05222164283874269855657485990, −6.34464177514932825075096971579, −4.62024962775250774887642551187, −3.92194037463887308749958368473, −2.03682061944970711169832779899, −0.69764404797797911279063758459, 2.00596795581338587519215709705, 3.16895884526786383208609758957, 3.94572575455594503554591472482, 5.92121359405180861821266600904, 6.51809371387861444932916360896, 7.78523504268503617995629499287, 8.419433520219889515897489085787, 9.402075052804518088803899002916, 10.04478479903165227013359297311, 11.04564944979398308132883505403

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