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Properties

Label 2-546-39.32-c1-0-5
Degree $2$
Conductor $546$
Sign $0.485 - 0.873i$
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  + (0.965 − 0.258i)2-s + (−1.39 + 1.02i)3-s + (0.866 − 0.499i)4-s + (2.04 + 2.04i)5-s + (−1.08 + 1.34i)6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (0.907 − 2.85i)9-s + (2.50 + 1.44i)10-s + (−0.446 − 1.66i)11-s + (−0.699 + 1.58i)12-s + (3.02 + 1.96i)13-s + i·14-s + (−4.94 − 0.766i)15-s + (0.500 − 0.866i)16-s + (3.47 + 6.02i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.807 + 0.590i)3-s + (0.433 − 0.249i)4-s + (0.913 + 0.913i)5-s + (−0.443 + 0.551i)6-s + (−0.0978 + 0.365i)7-s + (0.249 − 0.249i)8-s + (0.302 − 0.953i)9-s + (0.791 + 0.456i)10-s + (−0.134 − 0.502i)11-s + (−0.201 + 0.457i)12-s + (0.838 + 0.545i)13-s + 0.267i·14-s + (−1.27 − 0.197i)15-s + (0.125 − 0.216i)16-s + (0.842 + 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.485 - 0.873i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.485 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66161 + 0.977310i\)
\(L(\frac12)\) \(\approx\) \(1.66161 + 0.977310i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (1.39 - 1.02i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-3.02 - 1.96i)T \)
good5 \( 1 + (-2.04 - 2.04i)T + 5iT^{2} \)
11 \( 1 + (0.446 + 1.66i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.47 - 6.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.79 + 1.55i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.94 - 6.82i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.64 - 1.52i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.66 + 5.66i)T - 31iT^{2} \)
37 \( 1 + (5.75 - 1.54i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.05 - 0.817i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9.94 + 5.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.12 + 2.12i)T - 47iT^{2} \)
53 \( 1 + 1.27iT - 53T^{2} \)
59 \( 1 + (-2.54 - 0.682i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.57 + 4.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.346 - 1.29i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.86 + 10.7i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.64 + 6.64i)T + 73iT^{2} \)
79 \( 1 + 8.40T + 79T^{2} \)
83 \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \)
89 \( 1 + (0.857 + 3.20i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.03 + 2.15i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.81287757542459899034728122496, −10.41556683275318713590644122966, −9.557229925335915770829391228741, −8.382778197818023601209487389897, −6.79497287214261892014705740980, −5.99443412741338448186592468096, −5.73763755725614518062857295699, −4.22792160130478927395827874981, −3.32198209276861118915438998338, −1.84885477036989577946771028727, 1.08820285292195443904244860151, 2.48633767907486531366021303683, 4.34097293430060574986082196638, 5.14189039139890225618733094860, 5.98324011122807475417552816318, 6.69972785983802718517185897276, 7.81135164644319832583793431805, 8.727959029731354660038981984476, 10.10642327073423177078539638765, 10.59975417314602170776240709428

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