Properties

Label 2-4368-1.1-c1-0-1
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 − 5·11-s − 13-s + 15-s − 3·17-s + 19-s + 21-s − 3·23-s − 4·25-s − 27-s + 9·29-s − 4·31-s + 5·33-s + 35-s − 11·37-s + 39-s + 5·43-s − 45-s + 8·47-s + 49-s + 3·51-s − 2·53-s + 5·55-s − 57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.229·19-s + 0.218·21-s − 0.625·23-s − 4/5·25-s − 0.192·27-s + 1.67·29-s − 0.718·31-s + 0.870·33-s + 0.169·35-s − 1.80·37-s + 0.160·39-s + 0.762·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.420·51-s − 0.274·53-s + 0.674·55-s − 0.132·57-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: $\chi_{4368} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6121405627\)
\(L(\frac12)\) \(\approx\) \(0.6121405627\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.201446606753346861375494180077, −7.64539992819326394416926284918, −6.93384459729236425490988967620, −6.14870499157140837975820876453, −5.37589820595753831680253807256, −4.73726907840251180750454379627, −3.87503086378985783714020107268, −2.90477588733476870765553764796, −1.99947627461773626388751061444, −0.42907642254029688922261495011, 0.42907642254029688922261495011, 1.99947627461773626388751061444, 2.90477588733476870765553764796, 3.87503086378985783714020107268, 4.73726907840251180750454379627, 5.37589820595753831680253807256, 6.14870499157140837975820876453, 6.93384459729236425490988967620, 7.64539992819326394416926284918, 8.201446606753346861375494180077

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