↑

Properties

Label 2-1350-9.7-c1-0-4
Degree $2$
Conductor $1350$
Sign $0.800 - 0.598i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.18 + 2.05i)7-s − 0.999·8-s + (0.686 − 1.18i)11-s + (2.37 + 4.10i)13-s + (1.18 + 2.05i)14-s + (−0.5 + 0.866i)16-s − 7.37·17-s + 3.37·19-s + (−0.686 − 1.18i)22-s + (2.18 + 3.78i)23-s + 4.74·26-s + 2.37·28-s + (−2.18 + 3.78i)29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.448 + 0.776i)7-s − 0.353·8-s + (0.206 − 0.358i)11-s + (0.657 + 1.13i)13-s + (0.317 + 0.549i)14-s + (−0.125 + 0.216i)16-s − 1.78·17-s + 0.773·19-s + (−0.146 − 0.253i)22-s + (0.455 + 0.789i)23-s + 0.930·26-s + 0.448·28-s + (−0.405 + 0.703i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.800 - 0.598i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.800 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.518921737\)
\(L(\frac12)\) \(\approx\) \(1.518921737\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.18 - 2.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.686 + 1.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.37 - 4.10i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.37T + 17T^{2} \)
19 \( 1 - 3.37T + 19T^{2} \)
23 \( 1 + (-2.18 - 3.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.18 - 3.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.37 - 5.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.68 - 9.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.813 + 1.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + (0.686 + 1.18i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.55 - 7.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.813 - 1.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.11T + 89T^{2} \)
97 \( 1 + (1.31 - 2.27i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.477573478904409329434884160477, −9.116160604530539203097719117756, −8.397657390769558818145562194177, −6.96410904072018244418434872138, −6.37221030211350448392634298960, −5.43063259331106849398368286666, −4.49309436647894845531686685883, −3.52963761548160609499957729765, −2.57623469464808744662191450288, −1.42986992319072475202883683175, 0.57768095327140122740756497340, 2.47099230915486505267427229671, 3.66027291201539329773769919083, 4.37766815554762989460098229166, 5.38505988644591281667851504292, 6.36858332515990500959934303678, 6.93176679089196913227581720570, 7.81189701933564748354552819997, 8.576948804242984838731741582499, 9.452400304042564185944083478651

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