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Properties

Label 2-546-91.54-c1-0-12
Degree $2$
Conductor $546$
Sign $0.941 + 0.337i$
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  + (0.707 − 0.707i)2-s + (0.866 + 0.5i)3-s − 1.00i·4-s + (2.75 − 0.739i)5-s + (0.965 − 0.258i)6-s + (0.253 + 2.63i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (1.42 − 2.47i)10-s + (0.816 − 0.218i)11-s + (0.500 − 0.866i)12-s + (1.72 + 3.16i)13-s + (2.04 + 1.68i)14-s + (2.75 + 0.739i)15-s − 1.00·16-s − 5.69·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.499 + 0.288i)3-s − 0.500i·4-s + (1.23 − 0.330i)5-s + (0.394 − 0.105i)6-s + (0.0958 + 0.995i)7-s + (−0.250 − 0.250i)8-s + (0.166 + 0.288i)9-s + (0.451 − 0.781i)10-s + (0.246 − 0.0659i)11-s + (0.144 − 0.250i)12-s + (0.477 + 0.878i)13-s + (0.545 + 0.449i)14-s + (0.712 + 0.190i)15-s − 0.250·16-s − 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.941 + 0.337i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.941 + 0.337i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59359 - 0.450459i\)
\(L(\frac12)\) \(\approx\) \(2.59359 - 0.450459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.253 - 2.63i)T \)
13 \( 1 + (-1.72 - 3.16i)T \)
good5 \( 1 + (-2.75 + 0.739i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.816 + 0.218i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 5.69T + 17T^{2} \)
19 \( 1 + (-0.535 + 1.99i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.74iT - 23T^{2} \)
29 \( 1 + (1.07 + 1.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.489 - 1.82i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (5.26 + 5.26i)T + 37iT^{2} \)
41 \( 1 + (-2.29 + 8.55i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.409 + 0.236i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.98 - 11.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.82 + 6.62i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.99 + 4.99i)T - 59iT^{2} \)
61 \( 1 + (-6.83 + 3.94i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.942 - 3.51i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.65 - 9.90i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (15.3 + 4.11i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.97 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.53 + 1.53i)T + 83iT^{2} \)
89 \( 1 + (-7.75 + 7.75i)T - 89iT^{2} \)
97 \( 1 + (10.7 - 2.88i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.81842761644475647699122944729, −9.756417199682306619496225295634, −9.015231200497273737179892717745, −8.664465931223110847983468508483, −6.82963193872487702792775450308, −5.97751653291949433784250824250, −5.05776104316971712803369297393, −4.06418032444135298022074342958, −2.53890556095106439294064070800, −1.86135818948402190774173612831, 1.65727716763028151762487646103, 3.04200881422167733800419387490, 4.14030148872367299422687490347, 5.42765884642433594358680471187, 6.36095933824259682393075938396, 7.08175057816801123132114943003, 8.025572383898137503796640746671, 9.015437358268996635383191446606, 9.973871817018458841259639563709, 10.71663891257245696504738377550

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