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Properties

Label 2-1359-151.150-c0-0-9
Degree $2$
Conductor $1359$
Sign $1$
Analytic cond. $0.678229$
Root an. cond. $0.823546$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.80·2-s + 2.24·4-s + 0.445·5-s + 2.24·8-s + 0.801·10-s − 1.24·11-s + 1.80·16-s − 1.24·17-s − 1.80·19-s + 20-s − 2.24·22-s − 0.801·25-s + 1.80·29-s − 0.445·31-s + 1.00·32-s − 2.24·34-s + 1.24·37-s − 3.24·38-s + 1.00·40-s − 0.445·43-s − 2.80·44-s + 1.80·47-s + 49-s − 1.44·50-s − 0.554·55-s + 3.24·58-s + 0.445·59-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.24·4-s + 0.445·5-s + 2.24·8-s + 0.801·10-s − 1.24·11-s + 1.80·16-s − 1.24·17-s − 1.80·19-s + 20-s − 2.24·22-s − 0.801·25-s + 1.80·29-s − 0.445·31-s + 1.00·32-s − 2.24·34-s + 1.24·37-s − 3.24·38-s + 1.00·40-s − 0.445·43-s − 2.80·44-s + 1.80·47-s + 49-s − 1.44·50-s − 0.554·55-s + 3.24·58-s + 0.445·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1359\)    =    \(3^{2} \cdot 151\)
Sign: $1$
Analytic conductor: \(0.678229\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1359} (1207, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1359,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.929372830\)
\(L(\frac12)\) \(\approx\) \(2.929372830\)
\(L(1)\) 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 \( 1 - T \)
good2 \( 1 - 1.80T + T^{2} \)
5 \( 1 - 0.445T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.24T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.24T + T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.80T + T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 - 1.24T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 - 1.80T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 0.445T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.80T + T^{2} \)
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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

−10.24697744702199661033160404669, −8.894953056878616070937553644205, −7.968081098581943945174432235013, −6.93138443440063026344666257060, −6.24270944003378383400997589271, −5.56157638751936662955174099794, −4.62113793445675690106881207935, −4.06006335776010797616558749118, −2.65456490163878434562808350469, −2.20288923331080380611631481396, 2.20288923331080380611631481396, 2.65456490163878434562808350469, 4.06006335776010797616558749118, 4.62113793445675690106881207935, 5.56157638751936662955174099794, 6.24270944003378383400997589271, 6.93138443440063026344666257060, 7.968081098581943945174432235013, 8.894953056878616070937553644205, 10.24697744702199661033160404669

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