Properties

Label 2-546-91.74-c1-0-11
Degree $2$
Conductor $546$
Sign $0.658 - 0.752i$
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  + 2-s + (0.5 + 0.866i)3-s + 4-s + (1.14 + 1.98i)5-s + (0.5 + 0.866i)6-s + (2.63 + 0.222i)7-s + 8-s + (−0.499 + 0.866i)9-s + (1.14 + 1.98i)10-s + (−0.439 − 0.760i)11-s + (0.5 + 0.866i)12-s + (−0.786 − 3.51i)13-s + (2.63 + 0.222i)14-s + (−1.14 + 1.98i)15-s + 16-s − 6.40·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (0.512 + 0.887i)5-s + (0.204 + 0.353i)6-s + (0.996 + 0.0839i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.362 + 0.627i)10-s + (−0.132 − 0.229i)11-s + (0.144 + 0.249i)12-s + (−0.218 − 0.975i)13-s + (0.704 + 0.0593i)14-s + (−0.295 + 0.512i)15-s + 0.250·16-s − 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.658 - 0.752i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47685 + 1.12339i\)
\(L(\frac12)\) \(\approx\) \(2.47685 + 1.12339i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.63 - 0.222i)T \)
13 \( 1 + (0.786 + 3.51i)T \)
good5 \( 1 + (-1.14 - 1.98i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.439 + 0.760i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 6.40T + 17T^{2} \)
19 \( 1 + (0.754 - 1.30i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.31T + 23T^{2} \)
29 \( 1 + (0.669 - 1.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.94 - 3.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.38T + 37T^{2} \)
41 \( 1 + (-1.80 + 3.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.95 + 8.58i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.188 + 0.327i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.22 - 2.11i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.96T + 59T^{2} \)
61 \( 1 + (2.40 - 4.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.87 + 8.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.02 - 1.77i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.432 + 0.749i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.18 + 7.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.66T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + (-4.40 - 7.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.71187802316148297855280110009, −10.52015779372594292200147446809, −9.172198303940582763911651267515, −8.204634529147764646543003075093, −7.23701508558040268660710283976, −6.18369821772148924800770152927, −5.23814481514541132547737752018, −4.31328749991812180600579542339, −3.04025863020171689780142006450, −2.11600551289411832966245222616, 1.52684578173480360360349497596, 2.45696151963468978573467466366, 4.32745435039998707366762329490, 4.82574548881610371045279768163, 6.03180836615551235154425053326, 6.97702801821168664499023025556, 7.964921281867287381814842736287, 8.879698669091221033054551798024, 9.618769529670620949594762077921, 11.13064516753878519560473031149

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