Properties

Label 2-4368-1.1-c1-0-20
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 7-s + 9-s + 2·11-s + 13-s − 15-s − 4·17-s − 3·19-s − 21-s + 9·23-s − 4·25-s + 27-s − 29-s + 5·31-s + 2·33-s + 35-s − 8·37-s + 39-s + 6·41-s + 9·43-s − 45-s + 3·47-s + 49-s − 4·51-s + 3·53-s − 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.970·17-s − 0.688·19-s − 0.218·21-s + 1.87·23-s − 4/5·25-s + 0.192·27-s − 0.185·29-s + 0.898·31-s + 0.348·33-s + 0.169·35-s − 1.31·37-s + 0.160·39-s + 0.937·41-s + 1.37·43-s − 0.149·45-s + 0.437·47-s + 1/7·49-s − 0.560·51-s + 0.412·53-s − 0.269·55-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: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.087206603\)
\(L(\frac12)\) \(\approx\) \(2.087206603\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−8.485255636099615654192222300978, −7.65301309816855844423242627951, −6.90221224971830328003677088954, −6.41114902905702982840055909143, −5.37758607480975298731700050385, −4.36947028028289865835133717749, −3.85307545971783104916091823499, −2.94495441885535661130563296167, −2.06996964904661111806829340103, −0.791456854704856529223781893647, 0.791456854704856529223781893647, 2.06996964904661111806829340103, 2.94495441885535661130563296167, 3.85307545971783104916091823499, 4.36947028028289865835133717749, 5.37758607480975298731700050385, 6.41114902905702982840055909143, 6.90221224971830328003677088954, 7.65301309816855844423242627951, 8.485255636099615654192222300978

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