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Properties

Label 2-546-91.73-c1-0-6
Degree $2$
Conductor $546$
Sign $0.973 - 0.228i$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)3-s + (−0.866 − 0.499i)4-s + (−3.47 − 0.930i)5-s + (0.707 − 0.707i)6-s + (−1.68 + 2.03i)7-s + (0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (1.79 − 3.11i)10-s + (−0.175 − 0.653i)11-s + (0.5 + 0.866i)12-s + (2.65 + 2.44i)13-s + (−1.53 − 2.15i)14-s + (2.54 + 2.54i)15-s + (0.500 + 0.866i)16-s + (2.74 − 4.76i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.499 − 0.288i)3-s + (−0.433 − 0.249i)4-s + (−1.55 − 0.416i)5-s + (0.288 − 0.288i)6-s + (−0.636 + 0.770i)7-s + (0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (0.568 − 0.984i)10-s + (−0.0528 − 0.197i)11-s + (0.144 + 0.249i)12-s + (0.735 + 0.677i)13-s + (−0.409 − 0.576i)14-s + (0.656 + 0.656i)15-s + (0.125 + 0.216i)16-s + (0.666 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.973 - 0.228i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.973 - 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678468 + 0.0784417i\)
\(L(\frac12)\) \(\approx\) \(0.678468 + 0.0784417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (1.68 - 2.03i)T \)
13 \( 1 + (-2.65 - 2.44i)T \)
good5 \( 1 + (3.47 + 0.930i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.175 + 0.653i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.74 + 4.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0344 - 0.00922i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.63 + 3.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
31 \( 1 + (0.688 + 2.56i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.08 - 0.558i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.23 - 3.23i)T - 41iT^{2} \)
43 \( 1 - 6.92iT - 43T^{2} \)
47 \( 1 + (-2.41 + 8.99i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.982 - 1.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.70 - 1.52i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.80 + 2.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.22 - 1.39i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-9.32 - 9.32i)T + 71iT^{2} \)
73 \( 1 + (-3.38 + 0.906i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.22 - 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.11 - 1.11i)T - 83iT^{2} \)
89 \( 1 + (-1.55 + 5.80i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-11.0 + 11.0i)T - 97iT^{2} \)
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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

−11.08912998883378935155460602698, −9.746918406464742367578992306037, −8.791103495449685373064051651207, −8.173803996880529515681936137992, −7.16068279887743008993812096912, −6.45395198211733906412821261483, −5.29811978405292662825830207026, −4.39444315137879316355303721778, −3.11788435442905425429051735532, −0.70965094480007689985064841549, 0.862060256738173859618467888199, 3.31742236479569847891151182821, 3.69775149275797491902539647517, 4.82191184880868132890752597348, 6.26851185190866817280666992525, 7.34431418849204764210675084823, 8.048875740266503065448304935853, 9.109256477847383202894373246221, 10.44138717039614253237759284767, 10.60434450588820438192027154082

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