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Properties

Label 2-1359-151.67-c0-0-0
Degree $2$
Conductor $1359$
Sign $0.991 + 0.127i$
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 + (1.39 + 0.656i)7-s + (0.193 + 0.159i)13-s + (0.309 − 0.951i)16-s + (−1.41 + 1.03i)19-s + (−0.968 − 0.248i)25-s + (1.51 − 0.288i)28-s + (−0.620 + 0.582i)31-s + (−0.0800 + 0.0967i)37-s + (−0.791 − 1.68i)43-s + (0.876 + 1.05i)49-s + (0.250 + 0.0157i)52-s + (1.89 − 0.119i)61-s + (−0.309 − 0.951i)64-s + (−0.211 − 1.67i)67-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)4-s + (1.39 + 0.656i)7-s + (0.193 + 0.159i)13-s + (0.309 − 0.951i)16-s + (−1.41 + 1.03i)19-s + (−0.968 − 0.248i)25-s + (1.51 − 0.288i)28-s + (−0.620 + 0.582i)31-s + (−0.0800 + 0.0967i)37-s + (−0.791 − 1.68i)43-s + (0.876 + 1.05i)49-s + (0.250 + 0.0157i)52-s + (1.89 − 0.119i)61-s + (−0.309 − 0.951i)64-s + (−0.211 − 1.67i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1359\)    =    \(3^{2} \cdot 151\)
Sign: $0.991 + 0.127i$
Analytic conductor: \(0.678229\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1359} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1359,\ (\ :0),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.445778815\)
\(L(\frac12)\) \(\approx\) \(1.445778815\)
\(L(1)\) 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 \( 1 + (0.968 + 0.248i)T^{2} \)
7 \( 1 + (-1.39 - 0.656i)T + (0.637 + 0.770i)T^{2} \)
11 \( 1 + (-0.992 + 0.125i)T^{2} \)
13 \( 1 + (-0.193 - 0.159i)T + (0.187 + 0.982i)T^{2} \)
17 \( 1 + (-0.637 + 0.770i)T^{2} \)
19 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.728 - 0.684i)T^{2} \)
31 \( 1 + (0.620 - 0.582i)T + (0.0627 - 0.998i)T^{2} \)
37 \( 1 + (0.0800 - 0.0967i)T + (-0.187 - 0.982i)T^{2} \)
41 \( 1 + (-0.876 + 0.481i)T^{2} \)
43 \( 1 + (0.791 + 1.68i)T + (-0.637 + 0.770i)T^{2} \)
47 \( 1 + (0.876 - 0.481i)T^{2} \)
53 \( 1 + (0.425 + 0.904i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-1.89 + 0.119i)T + (0.992 - 0.125i)T^{2} \)
67 \( 1 + (0.211 + 1.67i)T + (-0.968 + 0.248i)T^{2} \)
71 \( 1 + (0.637 + 0.770i)T^{2} \)
73 \( 1 + (1.06 + 0.500i)T + (0.637 + 0.770i)T^{2} \)
79 \( 1 + (0.340 - 0.362i)T + (-0.0627 - 0.998i)T^{2} \)
83 \( 1 + (-0.535 - 0.844i)T^{2} \)
89 \( 1 + (-0.535 - 0.844i)T^{2} \)
97 \( 1 + (0.613 - 0.0774i)T + (0.968 - 0.248i)T^{2} \)
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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

−9.968683546979657899051627262025, −8.803641233266516937388172583611, −8.218837986556297148169306658746, −7.37007952801304796460611403482, −6.38318519759936337339773703378, −5.65212936549092956587385788959, −4.90266176234681937801062537030, −3.75077583081257742247288552228, −2.21320767726322418974504119334, −1.69124167382591436595166876114, 1.58615867058330829752218763986, 2.55041753588814234322334179244, 3.86821059220738662886961348118, 4.56929638276167001860429887349, 5.71305779246318352613194747188, 6.72104142000127683427013711929, 7.43179678357631206124489706360, 8.139544628192626196554216104413, 8.709117936462617447907550472229, 9.967843653299700995659327661419

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