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Properties

Label 2-1359-151.28-c0-0-0
Degree $2$
Conductor $1359$
Sign $0.514 - 0.857i$
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  + (0.809 + 0.587i)4-s + (0.0623 + 0.493i)7-s + (−0.488 + 1.90i)13-s + (0.309 + 0.951i)16-s + (−1.17 − 0.856i)19-s + (0.929 + 0.368i)25-s + (−0.239 + 0.435i)28-s + (0.844 − 1.79i)31-s + (−1.23 − 0.317i)37-s + (1.06 + 0.134i)43-s + (0.728 − 0.187i)49-s + (−1.51 + 1.25i)52-s + (−1.46 − 1.21i)61-s + (−0.309 + 0.951i)64-s + (1.96 − 0.374i)67-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)4-s + (0.0623 + 0.493i)7-s + (−0.488 + 1.90i)13-s + (0.309 + 0.951i)16-s + (−1.17 − 0.856i)19-s + (0.929 + 0.368i)25-s + (−0.239 + 0.435i)28-s + (0.844 − 1.79i)31-s + (−1.23 − 0.317i)37-s + (1.06 + 0.134i)43-s + (0.728 − 0.187i)49-s + (−1.51 + 1.25i)52-s + (−1.46 − 1.21i)61-s + (−0.309 + 0.951i)64-s + (1.96 − 0.374i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1359\)    =    \(3^{2} \cdot 151\)
Sign: $0.514 - 0.857i$
Analytic conductor: \(0.678229\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1359} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1359,\ (\ :0),\ 0.514 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.253579652\)
\(L(\frac12)\) \(\approx\) \(1.253579652\)
\(L(1)\) 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 \( 1 + (-0.929 - 0.368i)T^{2} \)
7 \( 1 + (-0.0623 - 0.493i)T + (-0.968 + 0.248i)T^{2} \)
11 \( 1 + (-0.187 - 0.982i)T^{2} \)
13 \( 1 + (0.488 - 1.90i)T + (-0.876 - 0.481i)T^{2} \)
17 \( 1 + (0.968 + 0.248i)T^{2} \)
19 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.425 + 0.904i)T^{2} \)
31 \( 1 + (-0.844 + 1.79i)T + (-0.637 - 0.770i)T^{2} \)
37 \( 1 + (1.23 + 0.317i)T + (0.876 + 0.481i)T^{2} \)
41 \( 1 + (-0.728 + 0.684i)T^{2} \)
43 \( 1 + (-1.06 - 0.134i)T + (0.968 + 0.248i)T^{2} \)
47 \( 1 + (0.728 - 0.684i)T^{2} \)
53 \( 1 + (0.992 + 0.125i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.46 + 1.21i)T + (0.187 + 0.982i)T^{2} \)
67 \( 1 + (-1.96 + 0.374i)T + (0.929 - 0.368i)T^{2} \)
71 \( 1 + (-0.968 + 0.248i)T^{2} \)
73 \( 1 + (-0.147 - 1.16i)T + (-0.968 + 0.248i)T^{2} \)
79 \( 1 + (0.666 - 0.313i)T + (0.637 - 0.770i)T^{2} \)
83 \( 1 + (-0.0627 - 0.998i)T^{2} \)
89 \( 1 + (-0.0627 - 0.998i)T^{2} \)
97 \( 1 + (0.115 + 0.607i)T + (-0.929 + 0.368i)T^{2} \)
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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

−9.832875586297268498795169386151, −8.999482416238119757754654632444, −8.414415879391617455439287742474, −7.31245534834991710146259885964, −6.76123856558234199619502640122, −6.02469472251913640786882559037, −4.72475503860755665012854888072, −3.94420454773218706116773978404, −2.60972453418247181074776542430, −1.96598632983747408463116708417, 1.10068769251878645030433556137, 2.48376988264601033302518915526, 3.38651526862895493201670720865, 4.74678897123829151220044924098, 5.54779823865720477320147011808, 6.39270735455490595568391947995, 7.18292575551355295934388537096, 7.968209196181954398935997093066, 8.762500214798425444280738958630, 10.03688801496672809024137861126

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