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Properties

Label 2-546-273.68-c1-0-17
Degree $2$
Conductor $546$
Sign $0.699 + 0.714i$
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  + i·2-s + (−0.926 − 1.46i)3-s − 4-s + (−2.14 + 3.71i)5-s + (1.46 − 0.926i)6-s + (−0.786 − 2.52i)7-s − i·8-s + (−1.28 + 2.71i)9-s + (−3.71 − 2.14i)10-s + (−0.940 − 0.542i)11-s + (0.926 + 1.46i)12-s + (2.47 − 2.62i)13-s + (2.52 − 0.786i)14-s + (7.43 − 0.301i)15-s + 16-s + 3.91·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.534 − 0.845i)3-s − 0.5·4-s + (−0.960 + 1.66i)5-s + (0.597 − 0.378i)6-s + (−0.297 − 0.954i)7-s − 0.353i·8-s + (−0.428 + 0.903i)9-s + (−1.17 − 0.679i)10-s + (−0.283 − 0.163i)11-s + (0.267 + 0.422i)12-s + (0.686 − 0.727i)13-s + (0.675 − 0.210i)14-s + (1.91 − 0.0777i)15-s + 0.250·16-s + 0.949·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.699 + 0.714i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629966 - 0.265049i\)
\(L(\frac12)\) \(\approx\) \(0.629966 - 0.265049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.926 + 1.46i)T \)
7 \( 1 + (0.786 + 2.52i)T \)
13 \( 1 + (-2.47 + 2.62i)T \)
good5 \( 1 + (2.14 - 3.71i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.940 + 0.542i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.91T + 17T^{2} \)
19 \( 1 + (-3.51 + 2.02i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.63iT - 23T^{2} \)
29 \( 1 + (1.84 - 1.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.04 - 0.606i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.07T + 37T^{2} \)
41 \( 1 + (-1.00 - 1.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.69 + 2.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.97 + 8.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.81 + 1.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.64T + 59T^{2} \)
61 \( 1 + (-12.0 + 6.96i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.01 - 6.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.68 + 1.55i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.86 + 2.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.33 + 4.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.52T + 83T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 + (-10.7 - 6.22i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.61307218175344881523809835082, −10.25379988782873157139669114507, −8.396740388462689698906768677031, −7.65643606098758642673589582453, −7.06114863282511079356454113287, −6.48029964723519354399006017631, −5.40482170438285880784236038153, −3.86421342521047588450528403174, −2.92430964035320518825172457375, −0.49057924172870938429468402097, 1.27726281660999463628646599658, 3.41272080504198954096575773194, 4.15152790220350426521897364482, 5.27369257563047038405730473905, 5.69230622034124417947573164967, 7.64079260339949350957794008444, 8.653608490694015842347869958287, 9.260634567179822146932291729850, 9.848983819009090710598835490909, 11.22498550072830176018509108507

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