↑

Properties

Label 2-1359-151.65-c0-0-0
Degree $2$
Conductor $1359$
Sign $0.181 + 0.983i$
Analytic cond. $0.678229$
Root an. cond. $0.823546$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)4-s + (−0.813 − 1.47i)7-s + (−1.15 + 0.733i)13-s + (0.309 − 0.951i)16-s + (1.60 − 1.16i)19-s + (−0.0627 − 0.998i)25-s + (−1.52 − 0.718i)28-s + (−0.328 + 1.72i)31-s + (−0.996 − 1.57i)37-s + (1.11 + 0.614i)43-s + (−0.992 + 1.56i)49-s + (−0.503 + 1.27i)52-s + (0.700 + 1.76i)61-s + (−0.309 − 0.951i)64-s + (−0.340 − 0.362i)67-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)4-s + (−0.813 − 1.47i)7-s + (−1.15 + 0.733i)13-s + (0.309 − 0.951i)16-s + (1.60 − 1.16i)19-s + (−0.0627 − 0.998i)25-s + (−1.52 − 0.718i)28-s + (−0.328 + 1.72i)31-s + (−0.996 − 1.57i)37-s + (1.11 + 0.614i)43-s + (−0.992 + 1.56i)49-s + (−0.503 + 1.27i)52-s + (0.700 + 1.76i)61-s + (−0.309 − 0.951i)64-s + (−0.340 − 0.362i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1359\)    =    \(3^{2} \cdot 151\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(0.678229\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1359} (820, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1359,\ (\ :0),\ 0.181 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.123303507\)
\(L(\frac12)\) \(\approx\) \(1.123303507\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
151 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.0627 + 0.998i)T^{2} \)
7 \( 1 + (0.813 + 1.47i)T + (-0.535 + 0.844i)T^{2} \)
11 \( 1 + (0.728 - 0.684i)T^{2} \)
13 \( 1 + (1.15 - 0.733i)T + (0.425 - 0.904i)T^{2} \)
17 \( 1 + (0.535 + 0.844i)T^{2} \)
19 \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.187 + 0.982i)T^{2} \)
31 \( 1 + (0.328 - 1.72i)T + (-0.929 - 0.368i)T^{2} \)
37 \( 1 + (0.996 + 1.57i)T + (-0.425 + 0.904i)T^{2} \)
41 \( 1 + (0.992 + 0.125i)T^{2} \)
43 \( 1 + (-1.11 - 0.614i)T + (0.535 + 0.844i)T^{2} \)
47 \( 1 + (-0.992 - 0.125i)T^{2} \)
53 \( 1 + (-0.876 - 0.481i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.700 - 1.76i)T + (-0.728 + 0.684i)T^{2} \)
67 \( 1 + (0.340 + 0.362i)T + (-0.0627 + 0.998i)T^{2} \)
71 \( 1 + (-0.535 + 0.844i)T^{2} \)
73 \( 1 + (-0.566 - 1.03i)T + (-0.535 + 0.844i)T^{2} \)
79 \( 1 + (-1.96 + 0.374i)T + (0.929 - 0.368i)T^{2} \)
83 \( 1 + (-0.968 + 0.248i)T^{2} \)
89 \( 1 + (-0.968 + 0.248i)T^{2} \)
97 \( 1 + (-0.450 + 0.423i)T + (0.0627 - 0.998i)T^{2} \)
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

−9.722236166849729865840197979636, −9.115231997582104119138502392065, −7.55843518109435283583677642357, −7.10347832159075107949673561066, −6.64135670380870186243091208691, −5.44630265722005580492925022417, −4.56393035254664510336435924234, −3.42138620458250580745931779431, −2.42222724783652749788920324649, −0.935818852222127903138219425790, 2.01144154386785541177519673621, 2.92704666797255324606957124767, 3.56910150123312470916906258545, 5.26294095515206424188779501058, 5.77739985853332350834739074587, 6.72245707792514083392739373726, 7.63716762638200513779239284391, 8.164018375919407036126791880148, 9.373894481447790219566895147720, 9.735506945000792036800587141054

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