↑

Properties

Label 2-726-11.5-c1-0-16
Degree $2$
Conductor $726$
Sign $0.998 - 0.0475i$
Analytic cond. $5.79713$
Root an. cond. $2.40772$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.618 − 1.90i)5-s + (−0.309 + 0.951i)6-s + (3.23 − 2.35i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 1.99·10-s − 0.999·12-s + (−1.85 − 5.70i)13-s + (3.23 + 2.35i)14-s + (1.61 − 1.17i)15-s + (0.309 − 0.951i)16-s + (0.618 − 1.90i)17-s + (−0.809 + 0.587i)18-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.276 − 0.850i)5-s + (−0.126 + 0.388i)6-s + (1.22 − 0.888i)7-s + (−0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + 0.632·10-s − 0.288·12-s + (−0.514 − 1.58i)13-s + (0.864 + 0.628i)14-s + (0.417 − 0.303i)15-s + (0.0772 − 0.237i)16-s + (0.149 − 0.461i)17-s + (−0.190 + 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.998 - 0.0475i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.998 - 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15023 + 0.0511291i\)
\(L(\frac12)\) \(\approx\) \(2.15023 + 0.0511291i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good5 \( 1 + (-0.618 + 1.90i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.23 + 2.35i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.85 + 5.70i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.618 + 1.90i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.23 + 2.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (4.85 - 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.85 - 3.52i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.85 - 3.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-9.70 - 7.05i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.618 - 1.90i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.70 - 7.05i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.32 - 13.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (3.70 - 11.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.85 + 3.52i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.23 + 3.80i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.23 + 3.80i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (4.32 + 13.3i)T + (-78.4 + 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.44005326493942870093576514655, −9.255147056907438520970748305102, −8.644087394867240012183465649604, −7.72011385322324271423416001974, −7.25905742616639216868164083608, −5.64888591909870714778711400255, −4.93363574505937219438927173571, −4.29725425886867408809502168007, −2.89017303423184125946070502076, −1.09362624168413256220891237558, 1.90317474129410795715809570706, 2.30506702065232659778556485691, 3.74210367070357818870859825927, 4.79588058163422112118269863629, 5.90916062499382798530107271668, 6.88768831346761832124626179714, 7.897669589760184348248762534902, 8.868285058204300678810877934409, 9.428578875693007048148042392573, 10.65756226569897940981182408748

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