Properties

Label 2-546-91.59-c1-0-9
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: $\chi_{546} (241, \cdot )$
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

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

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