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Properties

Label 2-546-273.269-c1-0-25
Degree $2$
Conductor $546$
Sign $0.970 - 0.239i$
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 + (0.144 + 1.72i)3-s − 4-s + (−1.41 − 2.44i)5-s + (−1.72 + 0.144i)6-s + (0.383 − 2.61i)7-s − i·8-s + (−2.95 + 0.498i)9-s + (2.44 − 1.41i)10-s + (4.64 − 2.68i)11-s + (−0.144 − 1.72i)12-s + (−2.04 + 2.97i)13-s + (2.61 + 0.383i)14-s + (4.02 − 2.79i)15-s + 16-s + 7.32·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.0833 + 0.996i)3-s − 0.5·4-s + (−0.632 − 1.09i)5-s + (−0.704 + 0.0589i)6-s + (0.145 − 0.989i)7-s − 0.353i·8-s + (−0.986 + 0.166i)9-s + (0.774 − 0.447i)10-s + (1.40 − 0.808i)11-s + (−0.0416 − 0.498i)12-s + (−0.565 + 0.824i)13-s + (0.699 + 0.102i)14-s + (1.03 − 0.721i)15-s + 0.250·16-s + 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25904 + 0.153095i\)
\(L(\frac12)\) \(\approx\) \(1.25904 + 0.153095i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.144 - 1.72i)T \)
7 \( 1 + (-0.383 + 2.61i)T \)
13 \( 1 + (2.04 - 2.97i)T \)
good5 \( 1 + (1.41 + 2.44i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.64 + 2.68i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 7.32T + 17T^{2} \)
19 \( 1 + (-2.17 - 1.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.84iT - 23T^{2} \)
29 \( 1 + (5.21 + 3.01i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.76 - 1.01i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.639T + 37T^{2} \)
41 \( 1 + (-5.97 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.836 + 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.48 - 4.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.82 + 1.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.67T + 59T^{2} \)
61 \( 1 + (-4.84 - 2.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.19 + 3.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.14 - 4.12i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.55 - 0.896i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.02 - 3.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + (4.57 - 2.64i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.71709900215061412807404269640, −9.660145011251222824587966712445, −9.079902020960056058243793476048, −8.215488981531979416274310651142, −7.42663712642632934002233989641, −6.12723308678385019141853878677, −5.08420094418937581692184118628, −4.16687236024662049845576067126, −3.65322515709720606550091474220, −0.849153476841683052605775738750, 1.46021714568124386589273883525, 2.81442485674998846043958723216, 3.51971820496621504571114569711, 5.24337236890404741485458740742, 6.25032140769544085221691660690, 7.42366953586596162415942528794, 7.83196679691481817469663225997, 9.163298165152041717658041214401, 9.844036916287157676665985051652, 11.11019819900985308556173460427

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