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Properties

Label 2-546-91.31-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.129 - 0.991i$
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 + (−0.535 + 1.99i)5-s + (0.707 + 0.707i)6-s + (−2.54 + 0.718i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.03 + 1.79i)10-s + (−1.83 + 0.490i)11-s + (0.5 + 0.866i)12-s + (2.43 + 2.65i)13-s + (−2.64 + 0.0351i)14-s + (−1.46 + 1.46i)15-s + (0.500 + 0.866i)16-s + (−1.24 + 2.16i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.499 + 0.288i)3-s + (0.433 + 0.249i)4-s + (−0.239 + 0.893i)5-s + (0.288 + 0.288i)6-s + (−0.962 + 0.271i)7-s + (0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.327 + 0.566i)10-s + (−0.552 + 0.148i)11-s + (0.144 + 0.249i)12-s + (0.676 + 0.736i)13-s + (−0.707 + 0.00938i)14-s + (−0.377 + 0.377i)15-s + (0.125 + 0.216i)16-s + (−0.302 + 0.524i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.129 - 0.991i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41420 + 1.61134i\)
\(L(\frac12)\) \(\approx\) \(1.41420 + 1.61134i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 - 0.718i)T \)
13 \( 1 + (-2.43 - 2.65i)T \)
good5 \( 1 + (0.535 - 1.99i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.83 - 0.490i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.24 - 2.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.191 + 0.714i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-7.35 + 4.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.10T + 29T^{2} \)
31 \( 1 + (-7.23 + 1.93i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.793 - 2.96i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.895 + 0.895i)T + 41iT^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (-1.80 - 0.482i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.70 + 4.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.48 - 12.9i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-8.23 + 4.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.30 + 8.60i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.51 - 4.51i)T - 71iT^{2} \)
73 \( 1 + (-0.185 - 0.693i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.82 - 8.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.63 + 1.63i)T + 83iT^{2} \)
89 \( 1 + (3.24 + 0.868i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.08 - 1.08i)T + 97iT^{2} \)
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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

−10.96115423034493311823656299034, −10.35109481111689585917828100179, −9.222997498102554391952665602127, −8.399374540094012609578379608516, −7.09884158023249557307781748433, −6.64739550854660232668426289509, −5.49209854600705492371580389559, −4.18500172127076787682762709441, −3.28622711454304387281003712023, −2.41150618130118851315006122929, 0.987007784765738752971015029891, 2.80646927296806739809508636769, 3.64334523592026068945247359340, 4.85890469289491122535434733235, 5.81836708899873756990948269133, 6.92169141115318751357705527052, 7.82687198450380967396904050476, 8.831578337984768454088787185806, 9.623253505730456449892707306527, 10.66066924472248129714307100139

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