Properties

Label 2-1359-151.83-c0-0-0
Degree $2$
Conductor $1359$
Sign $0.911 + 0.411i$
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 + (0.723 − 0.137i)7-s + (−0.354 + 0.895i)13-s + (0.309 − 0.951i)16-s + (0.688 − 0.500i)19-s + (−0.535 + 0.844i)25-s + (0.503 − 0.536i)28-s + (0.371 − 0.0469i)31-s + (−1.80 − 0.713i)37-s + (0.0235 − 0.123i)43-s + (−0.425 + 0.168i)49-s + (0.239 + 0.933i)52-s + (0.473 − 1.84i)61-s + (−0.309 − 0.951i)64-s + (−0.742 + 1.35i)67-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)4-s + (0.723 − 0.137i)7-s + (−0.354 + 0.895i)13-s + (0.309 − 0.951i)16-s + (0.688 − 0.500i)19-s + (−0.535 + 0.844i)25-s + (0.503 − 0.536i)28-s + (0.371 − 0.0469i)31-s + (−1.80 − 0.713i)37-s + (0.0235 − 0.123i)43-s + (−0.425 + 0.168i)49-s + (0.239 + 0.933i)52-s + (0.473 − 1.84i)61-s + (−0.309 − 0.951i)64-s + (−0.742 + 1.35i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1359\)    =    \(3^{2} \cdot 151\)
Sign: $0.911 + 0.411i$
Analytic conductor: \(0.678229\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1359} (838, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1359,\ (\ :0),\ 0.911 + 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.381667885\)
\(L(\frac12)\) \(\approx\) \(1.381667885\)
\(L(1)\) 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 \( 1 + (0.535 - 0.844i)T^{2} \)
7 \( 1 + (-0.723 + 0.137i)T + (0.929 - 0.368i)T^{2} \)
11 \( 1 + (0.876 + 0.481i)T^{2} \)
13 \( 1 + (0.354 - 0.895i)T + (-0.728 - 0.684i)T^{2} \)
17 \( 1 + (-0.929 - 0.368i)T^{2} \)
19 \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.992 + 0.125i)T^{2} \)
31 \( 1 + (-0.371 + 0.0469i)T + (0.968 - 0.248i)T^{2} \)
37 \( 1 + (1.80 + 0.713i)T + (0.728 + 0.684i)T^{2} \)
41 \( 1 + (0.425 - 0.904i)T^{2} \)
43 \( 1 + (-0.0235 + 0.123i)T + (-0.929 - 0.368i)T^{2} \)
47 \( 1 + (-0.425 + 0.904i)T^{2} \)
53 \( 1 + (0.187 - 0.982i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.473 + 1.84i)T + (-0.876 - 0.481i)T^{2} \)
67 \( 1 + (0.742 - 1.35i)T + (-0.535 - 0.844i)T^{2} \)
71 \( 1 + (0.929 - 0.368i)T^{2} \)
73 \( 1 + (-1.15 + 0.220i)T + (0.929 - 0.368i)T^{2} \)
79 \( 1 + (0.211 - 1.67i)T + (-0.968 - 0.248i)T^{2} \)
83 \( 1 + (0.637 - 0.770i)T^{2} \)
89 \( 1 + (0.637 - 0.770i)T^{2} \)
97 \( 1 + (-0.541 - 0.297i)T + (0.535 + 0.844i)T^{2} \)
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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

−9.752617857214869521927900949928, −9.079052976060389532403331164909, −7.980900166285651382065924083528, −7.19957570575898251393078015624, −6.60524142762001371117034880175, −5.48692973957101858577438739975, −4.88157594279102994258593751292, −3.63966612947502734096714829006, −2.35530393144434659934684250561, −1.43746327434114551256849676678, 1.64029634497934845396570303070, 2.74377732317314072203470304812, 3.64682365223368983205376484356, 4.83926497434524351426989615999, 5.73134134451491571684754020180, 6.65077620084797281601690396801, 7.59451461332997490039947452846, 8.056911095002830646349850278491, 8.839229853166184935102554090255, 10.09925777319010473264232834151

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