Properties

Label 2-381-1.1-c1-0-12
Degree $2$
Conductor $381$
Sign $-1$
Analytic cond. $3.04230$
Root an. cond. $1.74421$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.21·2-s − 3-s − 0.522·4-s + 1.83·5-s + 1.21·6-s − 2.18·7-s + 3.06·8-s + 9-s − 2.22·10-s − 4.11·11-s + 0.522·12-s + 2.89·13-s + 2.65·14-s − 1.83·15-s − 2.68·16-s − 2.75·17-s − 1.21·18-s + 4.55·19-s − 0.955·20-s + 2.18·21-s + 4.99·22-s − 8.48·23-s − 3.06·24-s − 1.64·25-s − 3.52·26-s − 27-s + 1.14·28-s + ⋯
L(s)  = 1  − 0.859·2-s − 0.577·3-s − 0.261·4-s + 0.818·5-s + 0.496·6-s − 0.825·7-s + 1.08·8-s + 0.333·9-s − 0.703·10-s − 1.23·11-s + 0.150·12-s + 0.803·13-s + 0.709·14-s − 0.472·15-s − 0.670·16-s − 0.668·17-s − 0.286·18-s + 1.04·19-s − 0.213·20-s + 0.476·21-s + 1.06·22-s − 1.76·23-s − 0.625·24-s − 0.329·25-s − 0.691·26-s − 0.192·27-s + 0.215·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(381\)    =    \(3 \cdot 127\)
Sign: $-1$
Analytic conductor: \(3.04230\)
Root analytic conductor: \(1.74421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 381,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
127 \( 1 + T \)
good2 \( 1 + 1.21T + 2T^{2} \)
5 \( 1 - 1.83T + 5T^{2} \)
7 \( 1 + 2.18T + 7T^{2} \)
11 \( 1 + 4.11T + 11T^{2} \)
13 \( 1 - 2.89T + 13T^{2} \)
17 \( 1 + 2.75T + 17T^{2} \)
19 \( 1 - 4.55T + 19T^{2} \)
23 \( 1 + 8.48T + 23T^{2} \)
29 \( 1 + 6.07T + 29T^{2} \)
31 \( 1 + 9.34T + 31T^{2} \)
37 \( 1 - 4.54T + 37T^{2} \)
41 \( 1 + 8.30T + 41T^{2} \)
43 \( 1 + 1.38T + 43T^{2} \)
47 \( 1 + 1.75T + 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 + 4.21T + 59T^{2} \)
61 \( 1 - 2.40T + 61T^{2} \)
67 \( 1 - 5.62T + 67T^{2} \)
71 \( 1 - 8.49T + 71T^{2} \)
73 \( 1 + 5.92T + 73T^{2} \)
79 \( 1 - 3.59T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 1.27T + 89T^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−10.57596627597327743246876334291, −9.907351094250031471994850267526, −9.328207685083987351826704743375, −8.196812200171199023402863754620, −7.24279852719451662190165713355, −6.00365731393032492292732215274, −5.27245644487825874142831058218, −3.77354264316338176067292941958, −1.91729121926422084500442340950, 0, 1.91729121926422084500442340950, 3.77354264316338176067292941958, 5.27245644487825874142831058218, 6.00365731393032492292732215274, 7.24279852719451662190165713355, 8.196812200171199023402863754620, 9.328207685083987351826704743375, 9.907351094250031471994850267526, 10.57596627597327743246876334291

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