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Properties

Label 2-1359-151.57-c0-0-0
Degree $2$
Conductor $1359$
Sign $0.150 - 0.988i$
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 + (0.503 + 1.27i)7-s + (1.15 − 1.23i)13-s + (−0.809 − 0.587i)16-s + (−0.393 + 1.21i)19-s + (0.425 + 0.904i)25-s + (−1.36 + 0.0859i)28-s + (−1.80 + 0.462i)31-s + (1.27 + 1.19i)37-s + (−1.84 − 0.730i)43-s + (−0.637 + 0.598i)49-s + (0.813 + 1.47i)52-s + (0.566 − 1.03i)61-s + (0.809 − 0.587i)64-s + (1.65 − 1.05i)67-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)4-s + (0.503 + 1.27i)7-s + (1.15 − 1.23i)13-s + (−0.809 − 0.587i)16-s + (−0.393 + 1.21i)19-s + (0.425 + 0.904i)25-s + (−1.36 + 0.0859i)28-s + (−1.80 + 0.462i)31-s + (1.27 + 1.19i)37-s + (−1.84 − 0.730i)43-s + (−0.637 + 0.598i)49-s + (0.813 + 1.47i)52-s + (0.566 − 1.03i)61-s + (0.809 − 0.587i)64-s + (1.65 − 1.05i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1359\)    =    \(3^{2} \cdot 151\)
Sign: $0.150 - 0.988i$
Analytic conductor: \(0.678229\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1359} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1359,\ (\ :0),\ 0.150 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.048636448\)
\(L(\frac12)\) \(\approx\) \(1.048636448\)
\(L(1)\) 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 \( 1 + (-0.425 - 0.904i)T^{2} \)
7 \( 1 + (-0.503 - 1.27i)T + (-0.728 + 0.684i)T^{2} \)
11 \( 1 + (0.535 + 0.844i)T^{2} \)
13 \( 1 + (-1.15 + 1.23i)T + (-0.0627 - 0.998i)T^{2} \)
17 \( 1 + (0.728 + 0.684i)T^{2} \)
19 \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (0.968 - 0.248i)T^{2} \)
31 \( 1 + (1.80 - 0.462i)T + (0.876 - 0.481i)T^{2} \)
37 \( 1 + (-1.27 - 1.19i)T + (0.0627 + 0.998i)T^{2} \)
41 \( 1 + (0.637 + 0.770i)T^{2} \)
43 \( 1 + (1.84 + 0.730i)T + (0.728 + 0.684i)T^{2} \)
47 \( 1 + (-0.637 - 0.770i)T^{2} \)
53 \( 1 + (0.929 + 0.368i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.566 + 1.03i)T + (-0.535 - 0.844i)T^{2} \)
67 \( 1 + (-1.65 + 1.05i)T + (0.425 - 0.904i)T^{2} \)
71 \( 1 + (-0.728 + 0.684i)T^{2} \)
73 \( 1 + (-0.700 - 1.76i)T + (-0.728 + 0.684i)T^{2} \)
79 \( 1 + (-0.450 + 1.75i)T + (-0.876 - 0.481i)T^{2} \)
83 \( 1 + (0.187 + 0.982i)T^{2} \)
89 \( 1 + (0.187 + 0.982i)T^{2} \)
97 \( 1 + (0.866 + 1.36i)T + (-0.425 + 0.904i)T^{2} \)
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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

−9.837619589069146926823730780506, −8.926845728994455211143148223458, −8.300171454880429392760733949738, −7.930150472817694616572356329657, −6.72111670020861565872959903320, −5.67053285255051223602853598049, −5.08426539608555124161460442068, −3.74342898562578890673110837796, −3.10273562841066834983885551513, −1.80698452316654623059925013661, 0.968393060239345825321019423699, 2.09882974744794936450499561840, 3.90130324945685428882534560356, 4.40368651258148753323400403724, 5.36179106131313752300098334825, 6.48211382321820732150167474385, 6.93787622829502075168977354000, 8.068688651984587130312327121345, 8.962721161568281854442553658383, 9.555161019566369132297675611933

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