↑

Properties

Label 2-1350-5.4-c3-0-25
Degree $2$
Conductor $1350$
Sign $0.894 + 0.447i$
Analytic cond. $79.6525$
Root an. cond. $8.92482$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 4·4-s + 15.1i·7-s + 8i·8-s − 34.3·11-s − 18.1i·13-s + 30.3·14-s + 16·16-s − 46.3i·17-s − 43.1·19-s + 68.7i·22-s − 20.7i·23-s − 36.3·26-s − 60.7i·28-s − 58.3·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.819i·7-s + 0.353i·8-s − 0.942·11-s − 0.387i·13-s + 0.579·14-s + 0.250·16-s − 0.661i·17-s − 0.521·19-s + 0.666i·22-s − 0.188i·23-s − 0.274·26-s − 0.409i·28-s − 0.373·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(79.6525\)
Root analytic conductor: \(8.92482\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.455083705\)
\(L(\frac12)\) \(\approx\) \(1.455083705\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 15.1iT - 343T^{2} \)
11 \( 1 + 34.3T + 1.33e3T^{2} \)
13 \( 1 + 18.1iT - 2.19e3T^{2} \)
17 \( 1 + 46.3iT - 4.91e3T^{2} \)
19 \( 1 + 43.1T + 6.85e3T^{2} \)
23 \( 1 + 20.7iT - 1.21e4T^{2} \)
29 \( 1 + 58.3T + 2.43e4T^{2} \)
31 \( 1 - 30.6T + 2.97e4T^{2} \)
37 \( 1 + 145. iT - 5.06e4T^{2} \)
41 \( 1 + 115.T + 6.89e4T^{2} \)
43 \( 1 - 317. iT - 7.95e4T^{2} \)
47 \( 1 + 375. iT - 1.03e5T^{2} \)
53 \( 1 - 119. iT - 1.48e5T^{2} \)
59 \( 1 + 161.T + 2.05e5T^{2} \)
61 \( 1 - 892.T + 2.26e5T^{2} \)
67 \( 1 - 289. iT - 3.00e5T^{2} \)
71 \( 1 - 702.T + 3.57e5T^{2} \)
73 \( 1 - 434. iT - 3.89e5T^{2} \)
79 \( 1 - 259T + 4.93e5T^{2} \)
83 \( 1 - 1.37e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.28e3T + 7.04e5T^{2} \)
97 \( 1 - 1.03e3iT - 9.12e5T^{2} \)
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

−9.262661976423010146175185680609, −8.470488231233713159420319543465, −7.79411203988984534401293035090, −6.68744996494522270937589766556, −5.55944797643248871226212631855, −5.04236349726292161060256175954, −3.87111062130187504674482989423, −2.77200899529916267051519021734, −2.14238144736018084255714003078, −0.63780643876065561056703729142, 0.54354537560786328102751311026, 1.97205607951920967532243044323, 3.39279885596293806597417642388, 4.29751972021202445658724727704, 5.13350729104360279913139149863, 6.08803064800938601901120605674, 6.89002603162395850839744315191, 7.65636596722198240645820038292, 8.308881585130705109420860440286, 9.164295514425174994294056978993

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