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Properties

Label 2-1359-151.3-c0-0-0
Degree $2$
Conductor $1359$
Sign $-0.423 + 0.905i$
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 + (−1.52 − 0.969i)7-s + (−1.80 − 0.849i)13-s + (−0.809 − 0.587i)16-s + (0.598 − 1.84i)19-s + (0.992 − 0.125i)25-s + (1.39 − 1.15i)28-s + (−0.996 − 0.394i)31-s + (−0.620 + 1.31i)37-s + (0.200 + 0.316i)43-s + (0.968 + 2.05i)49-s + (1.36 − 1.45i)52-s + (−0.804 − 0.856i)61-s + (0.809 − 0.587i)64-s + (−0.961 − 0.0604i)67-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)4-s + (−1.52 − 0.969i)7-s + (−1.80 − 0.849i)13-s + (−0.809 − 0.587i)16-s + (0.598 − 1.84i)19-s + (0.992 − 0.125i)25-s + (1.39 − 1.15i)28-s + (−0.996 − 0.394i)31-s + (−0.620 + 1.31i)37-s + (0.200 + 0.316i)43-s + (0.968 + 2.05i)49-s + (1.36 − 1.45i)52-s + (−0.804 − 0.856i)61-s + (0.809 − 0.587i)64-s + (−0.961 − 0.0604i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1359\)    =    \(3^{2} \cdot 151\)
Sign: $-0.423 + 0.905i$
Analytic conductor: \(0.678229\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1359} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1359,\ (\ :0),\ -0.423 + 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3720888305\)
\(L(\frac12)\) \(\approx\) \(0.3720888305\)
\(L(1)\) 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 \( 1 + (-0.992 + 0.125i)T^{2} \)
7 \( 1 + (1.52 + 0.969i)T + (0.425 + 0.904i)T^{2} \)
11 \( 1 + (0.0627 - 0.998i)T^{2} \)
13 \( 1 + (1.80 + 0.849i)T + (0.637 + 0.770i)T^{2} \)
17 \( 1 + (-0.425 + 0.904i)T^{2} \)
19 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.929 - 0.368i)T^{2} \)
31 \( 1 + (0.996 + 0.394i)T + (0.728 + 0.684i)T^{2} \)
37 \( 1 + (0.620 - 1.31i)T + (-0.637 - 0.770i)T^{2} \)
41 \( 1 + (-0.968 - 0.248i)T^{2} \)
43 \( 1 + (-0.200 - 0.316i)T + (-0.425 + 0.904i)T^{2} \)
47 \( 1 + (0.968 + 0.248i)T^{2} \)
53 \( 1 + (-0.535 - 0.844i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.804 + 0.856i)T + (-0.0627 + 0.998i)T^{2} \)
67 \( 1 + (0.961 + 0.0604i)T + (0.992 + 0.125i)T^{2} \)
71 \( 1 + (0.425 + 0.904i)T^{2} \)
73 \( 1 + (1.60 + 1.01i)T + (0.425 + 0.904i)T^{2} \)
79 \( 1 + (0.0922 + 0.233i)T + (-0.728 + 0.684i)T^{2} \)
83 \( 1 + (-0.876 + 0.481i)T^{2} \)
89 \( 1 + (-0.876 + 0.481i)T^{2} \)
97 \( 1 + (0.101 - 1.61i)T + (-0.992 - 0.125i)T^{2} \)
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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

−9.516517227617299216271099558449, −8.901093172191658977702974548453, −7.63366875169645946074087810629, −7.22729377914897647170227278771, −6.56337447142226022021427775152, −5.11951126524357043355600938226, −4.40062540888260953840717164081, −3.14593121098474926991620687087, −2.84256923553563081703864363779, −0.29219419956382558466585590418, 1.85919477169255140645299130696, 2.92696412584383235392782456563, 4.10483741309018872693506080790, 5.30273099197032989609302102146, 5.77749724962600605622694428442, 6.71556266371540971213321590923, 7.41724910143874025987785978388, 8.891778128956429349123674626465, 9.260819790624058071711008223189, 9.986919372739265264284368479801

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