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Properties

Label 2-1359-151.41-c0-0-0
Degree $2$
Conductor $1359$
Sign $0.998 - 0.0518i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)4-s + (1.51 − 1.25i)7-s + (0.488 − 0.0931i)13-s + (−0.809 − 0.587i)16-s + (0.331 − 1.01i)19-s + (−0.876 − 0.481i)25-s + (0.723 + 1.82i)28-s + (−0.0800 + 1.27i)31-s + (0.371 + 1.94i)37-s + (0.929 − 1.12i)43-s + (0.535 − 2.80i)49-s + (−0.0623 + 0.493i)52-s + (0.147 + 1.16i)61-s + (0.809 − 0.587i)64-s + (0.450 + 1.75i)67-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)4-s + (1.51 − 1.25i)7-s + (0.488 − 0.0931i)13-s + (−0.809 − 0.587i)16-s + (0.331 − 1.01i)19-s + (−0.876 − 0.481i)25-s + (0.723 + 1.82i)28-s + (−0.0800 + 1.27i)31-s + (0.371 + 1.94i)37-s + (0.929 − 1.12i)43-s + (0.535 − 2.80i)49-s + (−0.0623 + 0.493i)52-s + (0.147 + 1.16i)61-s + (0.809 − 0.587i)64-s + (0.450 + 1.75i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1359\)    =    \(3^{2} \cdot 151\)
Sign: $0.998 - 0.0518i$
Analytic conductor: \(0.678229\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1359} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1359,\ (\ :0),\ 0.998 - 0.0518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188649449\)
\(L(\frac12)\) \(\approx\) \(1.188649449\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.876 + 0.481i)T^{2} \)
7 \( 1 + (-1.51 + 1.25i)T + (0.187 - 0.982i)T^{2} \)
11 \( 1 + (0.968 - 0.248i)T^{2} \)
13 \( 1 + (-0.488 + 0.0931i)T + (0.929 - 0.368i)T^{2} \)
17 \( 1 + (-0.187 - 0.982i)T^{2} \)
19 \( 1 + (-0.331 + 1.01i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (0.0627 - 0.998i)T^{2} \)
31 \( 1 + (0.0800 - 1.27i)T + (-0.992 - 0.125i)T^{2} \)
37 \( 1 + (-0.371 - 1.94i)T + (-0.929 + 0.368i)T^{2} \)
41 \( 1 + (-0.535 + 0.844i)T^{2} \)
43 \( 1 + (-0.929 + 1.12i)T + (-0.187 - 0.982i)T^{2} \)
47 \( 1 + (0.535 - 0.844i)T^{2} \)
53 \( 1 + (0.637 - 0.770i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.147 - 1.16i)T + (-0.968 + 0.248i)T^{2} \)
67 \( 1 + (-0.450 - 1.75i)T + (-0.876 + 0.481i)T^{2} \)
71 \( 1 + (0.187 - 0.982i)T^{2} \)
73 \( 1 + (1.46 - 1.21i)T + (0.187 - 0.982i)T^{2} \)
79 \( 1 + (0.961 - 0.0604i)T + (0.992 - 0.125i)T^{2} \)
83 \( 1 + (0.425 - 0.904i)T^{2} \)
89 \( 1 + (0.425 - 0.904i)T^{2} \)
97 \( 1 + (1.56 - 0.402i)T + (0.876 - 0.481i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.858039834990268668658544438822, −8.604741474478298375681336581503, −8.323054983503946231753946848895, −7.36485126417524396719412590853, −6.92801441913627470065615431593, −5.41437713233387970817737431559, −4.49580199872147250547590179565, −3.98450452550109353111804591283, −2.77877687499803215900266453726, −1.27816855089990146138542744716, 1.50034659661583888465841534006, 2.28159260847323008345206336077, 3.96425063431441283526959664382, 4.87352112535878502048658125129, 5.75134548368489104944554286820, 6.02436739473570704167447494746, 7.61148328736389096190117281241, 8.180901823197696678361957233582, 9.147501534306397979840116153694, 9.544943638220746388507335007222

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