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Properties

Label 2-546-39.32-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.452 - 0.891i$
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 + (−0.847 + 1.51i)3-s + (0.866 − 0.499i)4-s + (1.36 + 1.36i)5-s + (0.427 − 1.67i)6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (−1.56 − 2.55i)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 + (−3.22 + 0.908i)15-s + (0.500 − 0.866i)16-s + (0.540 + 0.935i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.489 + 0.872i)3-s + (0.433 − 0.249i)4-s + (0.612 + 0.612i)5-s + (0.174 − 0.685i)6-s + (0.0978 − 0.365i)7-s + (−0.249 + 0.249i)8-s + (−0.521 − 0.853i)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 + (−0.833 + 0.234i)15-s + (0.125 − 0.216i)16-s + (0.131 + 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.452 - 0.891i$
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: \((2,\ 546,\ (\ :1/2),\ -0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507093 + 0.825702i\)
\(L(\frac12)\) \(\approx\) \(0.507093 + 0.825702i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.847 - 1.51i)T \)
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

−10.80425790421799702317092161337, −10.09295841984148105459508388272, −9.613993680364793794571504771962, −8.661176024882927614203453295566, −7.40177438085572229162917026864, −6.53906448086425780657166473778, −5.72731040152044623448254404916, −4.52697692184073063349947600734, −3.33540741736637415166580736937, −1.63825430107867133246503078451, 0.796111130707311506862708659924, 1.92366689010391526790976405605, 3.41084777840330841955520519857, 5.31197516058978785666788096405, 5.96754794794742610876172112850, 6.83673843806023268063900272971, 8.113264086815603188160990958298, 8.637337745616596358791813500473, 9.473086745975299175808342990112, 10.69382856288873543188619141208

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