↑

Properties

Label 2-1350-45.34-c1-0-16
Degree $2$
Conductor $1350$
Sign $-0.980 + 0.195i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (0.389 + 0.224i)7-s − 0.999i·8-s + (−2.44 + 4.24i)11-s + (0.389 − 0.224i)13-s + (−0.224 − 0.389i)14-s + (−0.5 + 0.866i)16-s − 4.89i·17-s − 7.44·19-s + (4.24 − 2.44i)22-s + (−2.12 + 1.22i)23-s − 0.449·26-s + 0.449i·28-s + (1.22 − 2.12i)29-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.147 + 0.0849i)7-s − 0.353i·8-s + (−0.738 + 1.27i)11-s + (0.107 − 0.0623i)13-s + (−0.0600 − 0.104i)14-s + (−0.125 + 0.216i)16-s − 1.18i·17-s − 1.70·19-s + (0.904 − 0.522i)22-s + (−0.442 + 0.255i)23-s − 0.0881·26-s + 0.0849i·28-s + (0.227 − 0.393i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.980 + 0.195i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2130683417\)
\(L(\frac12)\) \(\approx\) \(0.2130683417\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.389 - 0.224i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.44 - 4.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.389 + 0.224i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 + (2.12 - 1.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 + 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.22 + 3.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.20 - 1.27i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.43 + 5.44i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.55iT - 53T^{2} \)
59 \( 1 + (-2.72 - 4.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.301 + 0.174i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - iT - 73T^{2} \)
79 \( 1 + (-8.34 + 14.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.71 - 2.72i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (7.61 + 4.39i)T + (48.5 + 84.0i)T^{2} \)
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

−9.373442301791414041863858157950, −8.453505232763313032449521991846, −7.67200150899748574084827668781, −7.03425187799792393197020344952, −5.99030135531057856854626751787, −4.85561133051351852743234338798, −4.05124396753137062726543635823, −2.64111953416455343705167776431, −1.91121521712688236787892436783, −0.10246353154306206935961105424, 1.51733003721263135997844622854, 2.80411736353517922731003428963, 3.99685387688611163999115370024, 5.11141767083487073126422444153, 6.12498154022965176259003090909, 6.56416748692617554562181564673, 7.86835728255680159585753695233, 8.339490578109850870881137555081, 8.917825423181372554618526657774, 10.04182254225589312621368954788

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