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Properties

Label 2-546-39.32-c1-0-18
Degree $2$
Conductor $546$
Sign $0.429 + 0.903i$
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 + (−0.884 + 1.48i)3-s + (0.866 − 0.499i)4-s + (−1.36 − 1.36i)5-s + (−0.469 + 1.66i)6-s + (0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (−1.43 − 2.63i)9-s + (−1.67 − 0.967i)10-s + (−1.29 − 4.83i)11-s + (−0.0218 + 1.73i)12-s + (3.60 + 0.102i)13-s − i·14-s + (3.24 − 0.826i)15-s + (0.500 − 0.866i)16-s + (−0.540 − 0.935i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.510 + 0.859i)3-s + (0.433 − 0.249i)4-s + (−0.612 − 0.612i)5-s + (−0.191 + 0.680i)6-s + (0.0978 − 0.365i)7-s + (0.249 − 0.249i)8-s + (−0.477 − 0.878i)9-s + (−0.530 − 0.306i)10-s + (−0.390 − 1.45i)11-s + (−0.00631 + 0.499i)12-s + (0.999 + 0.0283i)13-s − 0.267i·14-s + (0.839 − 0.213i)15-s + (0.125 − 0.216i)16-s + (−0.131 − 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.429 + 0.903i$
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: \((2,\ 546,\ (\ :1/2),\ 0.429 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28238 - 0.810046i\)
\(L(\frac12)\) \(\approx\) \(1.28238 - 0.810046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.884 - 1.48i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-3.60 - 0.102i)T \)
good5 \( 1 + (1.36 + 1.36i)T + 5iT^{2} \)
11 \( 1 + (1.29 + 4.83i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.540 + 0.935i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.27 - 0.609i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.57 + 2.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.710 - 0.410i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.20 - 4.20i)T - 31iT^{2} \)
37 \( 1 + (-0.524 + 0.140i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.18 + 1.38i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.92 - 2.84i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.118 - 0.118i)T - 47iT^{2} \)
53 \( 1 + 14.4iT - 53T^{2} \)
59 \( 1 + (1.65 + 0.444i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.49 + 7.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.08 - 11.5i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.61 - 9.74i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.51 - 1.51i)T + 73iT^{2} \)
79 \( 1 + 1.54T + 79T^{2} \)
83 \( 1 + (0.338 + 0.338i)T + 83iT^{2} \)
89 \( 1 + (-4.35 - 16.2i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.07 - 1.09i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.05191119314039831601969398988, −10.01713106959505681351923916252, −8.819637807754300208524164556314, −8.176270977258171954724641214335, −6.70603084906749105942469689557, −5.73165216609213009235556811523, −4.93959548830809973431894072881, −3.93676168898469560246807775000, −3.19376239426849612790585464427, −0.78320008116808999417281417959, 1.81171116840333019587575347525, 3.10467925466332689806264789649, 4.43750274367240929638374904245, 5.51296090312916271122387002862, 6.38726907406034935017045030341, 7.40565174226433245795728856789, 7.70707471530897700315772719550, 9.073565008601752722287485769738, 10.47319100084023686125597313667, 11.23015065740081428847127366317

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