Properties

Label 2-4368-1.1-c1-0-51
Degree $2$
Conductor $4368$
Sign $-1$
Analytic cond. $34.8786$
Root an. cond. $5.90581$
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  − 3-s + 2·5-s − 7-s + 9-s − 4·11-s − 13-s − 2·15-s + 6·17-s − 4·19-s + 21-s + 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s + 4·33-s − 2·35-s + 2·37-s + 39-s + 2·41-s + 4·43-s + 2·45-s + 12·47-s + 49-s − 6·51-s − 6·53-s − 8·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.516·15-s + 1.45·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.338·35-s + 0.328·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(34.8786\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ -1)\)

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

−7.66467345457669052080920033437, −7.44270336277053678144713883051, −6.37607503793157208580876606764, −5.61229191224670573567150649200, −5.38346092555721800780730682119, −4.35723405363722063099275405612, −3.24931967921157499778862027219, −2.42938437204153413511200868134, −1.38065412823168408113556456323, 0, 1.38065412823168408113556456323, 2.42938437204153413511200868134, 3.24931967921157499778862027219, 4.35723405363722063099275405612, 5.38346092555721800780730682119, 5.61229191224670573567150649200, 6.37607503793157208580876606764, 7.44270336277053678144713883051, 7.66467345457669052080920033437

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