Properties

Label 2-1350-9.7-c1-0-11
Degree $2$
Conductor $1350$
Sign $-0.173 + 0.984i$
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 + (−2 + 3.46i)7-s − 0.999·8-s + (1.5 − 2.59i)11-s + (−2 − 3.46i)13-s + (1.99 + 3.46i)14-s + (−0.5 + 0.866i)16-s + 3·17-s + 5·19-s + (−1.5 − 2.59i)22-s + (−3 − 5.19i)23-s − 3.99·26-s + 3.99·28-s + (3 − 5.19i)29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.755 + 1.30i)7-s − 0.353·8-s + (0.452 − 0.783i)11-s + (−0.554 − 0.960i)13-s + (0.534 + 0.925i)14-s + (−0.125 + 0.216i)16-s + 0.727·17-s + 1.14·19-s + (−0.319 − 0.553i)22-s + (−0.625 − 1.08i)23-s − 0.784·26-s + 0.755·28-s + (0.557 − 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.173 + 0.984i$
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: \((2,\ 1350,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.602130199\)
\(L(\frac12)\) \(\approx\) \(1.602130199\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-5.5 + 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.605848215609773402939544957879, −8.691404646559196459173988024168, −7.967605571141222859528505860505, −6.71248658222267425331884506450, −5.69838773170041315387570530160, −5.45702526952035240486692853150, −4.01939987469260157825986653599, −3.04061324049766734339289166861, −2.40078938437342417576423649212, −0.64604342364778283633835676747, 1.30698843814315406268481776280, 3.05693675681021222101574531057, 3.95731197952804616037349888435, 4.66996691897144852334544346830, 5.74680716443291661824498655398, 6.76619595709434143554395487968, 7.24627048369732391999283594904, 7.84144896634660928446929687852, 9.215091751176277948806727407592, 9.699351818900393595822257050111

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