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Properties

Label 2-546-91.5-c1-0-2
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: $\chi_{546} (187, \cdot )$
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

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

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