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Properties

Label 2-1359-151.107-c0-0-0
Degree $2$
Conductor $1359$
Sign $-0.248 + 0.968i$
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 + (−0.250 − 0.0157i)7-s + (−0.193 − 1.52i)13-s + (−0.809 + 0.587i)16-s + (−0.574 − 1.76i)19-s + (0.187 + 0.982i)25-s + (0.0623 + 0.242i)28-s + (0.0672 − 0.106i)31-s + (0.844 − 0.106i)37-s + (−0.110 − 1.74i)43-s + (−0.929 − 0.117i)49-s + (−1.39 + 0.656i)52-s + (−1.06 − 0.500i)61-s + (0.809 + 0.587i)64-s + (−1.05 + 0.872i)67-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)4-s + (−0.250 − 0.0157i)7-s + (−0.193 − 1.52i)13-s + (−0.809 + 0.587i)16-s + (−0.574 − 1.76i)19-s + (0.187 + 0.982i)25-s + (0.0623 + 0.242i)28-s + (0.0672 − 0.106i)31-s + (0.844 − 0.106i)37-s + (−0.110 − 1.74i)43-s + (−0.929 − 0.117i)49-s + (−1.39 + 0.656i)52-s + (−1.06 − 0.500i)61-s + (0.809 + 0.587i)64-s + (−1.05 + 0.872i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1359\)    =    \(3^{2} \cdot 151\)
Sign: $-0.248 + 0.968i$
Analytic conductor: \(0.678229\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1359} (1315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1359,\ (\ :0),\ -0.248 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8254250131\)
\(L(\frac12)\) \(\approx\) \(0.8254250131\)
\(L(1)\) 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 \( 1 + (-0.187 - 0.982i)T^{2} \)
7 \( 1 + (0.250 + 0.0157i)T + (0.992 + 0.125i)T^{2} \)
11 \( 1 + (-0.637 - 0.770i)T^{2} \)
13 \( 1 + (0.193 + 1.52i)T + (-0.968 + 0.248i)T^{2} \)
17 \( 1 + (-0.992 + 0.125i)T^{2} \)
19 \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.535 - 0.844i)T^{2} \)
31 \( 1 + (-0.0672 + 0.106i)T + (-0.425 - 0.904i)T^{2} \)
37 \( 1 + (-0.844 + 0.106i)T + (0.968 - 0.248i)T^{2} \)
41 \( 1 + (0.929 + 0.368i)T^{2} \)
43 \( 1 + (0.110 + 1.74i)T + (-0.992 + 0.125i)T^{2} \)
47 \( 1 + (-0.929 - 0.368i)T^{2} \)
53 \( 1 + (-0.0627 - 0.998i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.06 + 0.500i)T + (0.637 + 0.770i)T^{2} \)
67 \( 1 + (1.05 - 0.872i)T + (0.187 - 0.982i)T^{2} \)
71 \( 1 + (0.992 + 0.125i)T^{2} \)
73 \( 1 + (-1.89 - 0.119i)T + (0.992 + 0.125i)T^{2} \)
79 \( 1 + (-1.65 + 1.05i)T + (0.425 - 0.904i)T^{2} \)
83 \( 1 + (-0.728 + 0.684i)T^{2} \)
89 \( 1 + (-0.728 + 0.684i)T^{2} \)
97 \( 1 + (-1.03 - 1.24i)T + (-0.187 + 0.982i)T^{2} \)
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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

−9.481620871804872664609391832255, −8.988274487861833557588711257215, −7.980882485121892343918176797930, −7.02534930937384347192915882500, −6.19186030031122433653774677851, −5.31283496301083673364224427255, −4.70829191303662242025804288034, −3.40700592390400590847168496788, −2.28155916331868119028007414349, −0.69377189933055202663425014010, 1.90123932635303870706050548562, 3.09316372072738550827532558328, 4.10793703457969832699634042429, 4.66597862951501127101422080863, 6.11728619025862373114924759788, 6.72336372121208835461017274258, 7.80351143197116217514299429497, 8.295626139604171012511102882319, 9.260852695202845464181196817569, 9.792130405365165697692064068530

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