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Properties

Label 2-546-91.54-c1-0-7
Degree $2$
Conductor $546$
Sign $0.974 - 0.222i$
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 + (3.01 − 0.807i)5-s + (0.965 − 0.258i)6-s + (2.56 + 0.633i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.55 + 2.70i)10-s + (−1.47 + 0.396i)11-s + (−0.500 + 0.866i)12-s + (−0.598 + 3.55i)13-s + (−2.26 + 1.36i)14-s + (−3.01 − 0.807i)15-s − 1.00·16-s + 3.46·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.499 − 0.288i)3-s − 0.500i·4-s + (1.34 − 0.361i)5-s + (0.394 − 0.105i)6-s + (0.970 + 0.239i)7-s + (0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (−0.493 + 0.854i)10-s + (−0.445 + 0.119i)11-s + (−0.144 + 0.250i)12-s + (−0.165 + 0.986i)13-s + (−0.605 + 0.365i)14-s + (−0.777 − 0.208i)15-s − 0.250·16-s + 0.840·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.974 - 0.222i$
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: \((2,\ 546,\ (\ :1/2),\ 0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32067 + 0.148918i\)
\(L(\frac12)\) \(\approx\) \(1.32067 + 0.148918i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 0.633i)T \)
13 \( 1 + (0.598 - 3.55i)T \)
good5 \( 1 + (-3.01 + 0.807i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.47 - 0.396i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (-1.28 + 4.81i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 7.42iT - 23T^{2} \)
29 \( 1 + (-1.30 - 2.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.72 - 10.1i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.34 - 3.34i)T + 37iT^{2} \)
41 \( 1 + (0.819 - 3.05i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.51 - 3.76i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.52 + 9.42i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.34 + 5.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.02 + 6.02i)T - 59iT^{2} \)
61 \( 1 + (-6.60 + 3.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.779 - 2.90i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.689 - 2.57i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.59 + 1.76i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.33 + 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.39 + 5.39i)T + 83iT^{2} \)
89 \( 1 + (5.62 - 5.62i)T - 89iT^{2} \)
97 \( 1 + (4.27 - 1.14i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.67173141577753502173671403919, −9.905044240500382215957876754036, −9.002081697568729468577984096193, −8.277228300204517009363482868500, −7.08267373899705893615645720418, −6.34915263957752736932870856022, −5.22182653472779471348146706912, −4.84567899529019475380737242185, −2.36431612958199536528974458130, −1.29115808017434271608929649591, 1.29193682760715341291959833817, 2.54781748277077383463750641511, 3.94953339190396612527707387822, 5.54577110637704975103505918966, 5.74366728657255026822584383185, 7.45079649740701706104826114301, 8.022354506400427553634960749660, 9.459875054942197987667386019001, 9.942319148361223266378318577947, 10.66898839819579565442012887303

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