Properties

Label 2-234-1.1-c1-0-4
Degree $2$
Conductor $234$
Sign $-1$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s − 2·7-s − 8-s + 2·10-s − 4·11-s − 13-s + 2·14-s + 16-s − 6·19-s − 2·20-s + 4·22-s + 4·23-s − 25-s + 26-s − 2·28-s − 8·29-s − 2·31-s − 32-s + 4·35-s + 6·37-s + 6·38-s + 2·40-s + 6·41-s − 8·43-s − 4·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.37·19-s − 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.377·28-s − 1.48·29-s − 0.359·31-s − 0.176·32-s + 0.676·35-s + 0.986·37-s + 0.973·38-s + 0.316·40-s + 0.937·41-s − 1.21·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 \)
13 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−11.48471457324402872742962494510, −10.70725444808913514749917450090, −9.773699703719295926726447008045, −8.695739660347841822203445826112, −7.76516545555292037952264969735, −6.93707785293490008685942511031, −5.59584593532556316161391417712, −3.99394475360161838020150844401, −2.57523735650192716532994366231, 0, 2.57523735650192716532994366231, 3.99394475360161838020150844401, 5.59584593532556316161391417712, 6.93707785293490008685942511031, 7.76516545555292037952264969735, 8.695739660347841822203445826112, 9.773699703719295926726447008045, 10.70725444808913514749917450090, 11.48471457324402872742962494510

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