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Properties

Label 2-546-39.32-c1-0-22
Degree $2$
Conductor $546$
Sign $-0.578 + 0.815i$
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 + (−1.67 + 0.438i)3-s + (0.866 − 0.499i)4-s + (−1.24 − 1.24i)5-s + (−1.50 + 0.856i)6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (2.61 − 1.46i)9-s + (−1.52 − 0.879i)10-s + (−0.417 − 1.55i)11-s + (−1.23 + 1.21i)12-s + (−3.43 − 1.10i)13-s + i·14-s + (2.62 + 1.53i)15-s + (0.500 − 0.866i)16-s + (−2.60 − 4.50i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.967 + 0.252i)3-s + (0.433 − 0.249i)4-s + (−0.556 − 0.556i)5-s + (−0.614 + 0.349i)6-s + (−0.0978 + 0.365i)7-s + (0.249 − 0.249i)8-s + (0.871 − 0.489i)9-s + (−0.481 − 0.278i)10-s + (−0.125 − 0.470i)11-s + (−0.355 + 0.351i)12-s + (−0.952 − 0.305i)13-s + 0.267i·14-s + (0.679 + 0.397i)15-s + (0.125 − 0.216i)16-s + (−0.631 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.578 + 0.815i$
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: \((2,\ 546,\ (\ :1/2),\ -0.578 + 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409027 - 0.791670i\)
\(L(\frac12)\) \(\approx\) \(0.409027 - 0.791670i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 0.438i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (3.43 + 1.10i)T \)
good5 \( 1 + (1.24 + 1.24i)T + 5iT^{2} \)
11 \( 1 + (0.417 + 1.55i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.60 + 4.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.82 + 0.490i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.39 + 2.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.73 + 3.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.89 + 1.89i)T - 31iT^{2} \)
37 \( 1 + (0.216 - 0.0580i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.320 - 0.0859i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.90 - 1.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.55 + 5.55i)T - 47iT^{2} \)
53 \( 1 - 6.37iT - 53T^{2} \)
59 \( 1 + (-5.66 - 1.51i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.51 - 6.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.05 - 7.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.213 + 0.796i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.64 + 4.64i)T + 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + (-0.280 - 0.280i)T + 83iT^{2} \)
89 \( 1 + (2.21 + 8.25i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.03 + 0.276i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.73162688753856984445484651039, −9.821474429610050152068870685113, −8.841624705757932161498432985253, −7.58675012080270241522452870169, −6.64516459390892905947698485529, −5.60423513956384961152370447767, −4.84416250582498894734291603122, −4.04651341606946877732486557870, −2.53452248463875162469675662272, −0.44553242390896455203156152494, 1.97355657246804645925545338077, 3.63717856615224499911218264994, 4.57399860566781777416559560446, 5.52571803517844608347247810246, 6.69757561445027841569147005810, 7.13276020501665640310071075227, 8.034111378518061994587087347066, 9.580443327388940593758153178383, 10.64269982524525038013684168363, 11.12162559518682375679202004790

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