Properties

Label 2-5408-1.1-c1-0-99
Degree $2$
Conductor $5408$
Sign $1$
Analytic cond. $43.1830$
Root an. cond. $6.57138$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.96·3-s + 1.23·5-s + 4.80·7-s + 5.80·9-s − 4.80·11-s + 3.66·15-s − 17-s + 5.13·19-s + 14.2·21-s − 3.96·23-s − 3.47·25-s + 8.32·27-s + 1.33·29-s + 3.66·31-s − 14.2·33-s + 5.93·35-s − 3.62·37-s + 7.66·41-s − 2.38·43-s + 7.16·45-s + 5.94·47-s + 16.0·49-s − 2.96·51-s + 0.139·53-s − 5.93·55-s + 15.2·57-s + 5.47·59-s + ⋯
L(s)  = 1  + 1.71·3-s + 0.552·5-s + 1.81·7-s + 1.93·9-s − 1.44·11-s + 0.946·15-s − 0.242·17-s + 1.17·19-s + 3.11·21-s − 0.825·23-s − 0.694·25-s + 1.60·27-s + 0.247·29-s + 0.658·31-s − 2.48·33-s + 1.00·35-s − 0.595·37-s + 1.19·41-s − 0.364·43-s + 1.06·45-s + 0.867·47-s + 2.29·49-s − 0.415·51-s + 0.0191·53-s − 0.800·55-s + 2.01·57-s + 0.712·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5408\)    =    \(2^{5} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(43.1830\)
Root analytic conductor: \(6.57138\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.137554272\)
\(L(\frac12)\) \(\approx\) \(5.137554272\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 - 2.96T + 3T^{2} \)
5 \( 1 - 1.23T + 5T^{2} \)
7 \( 1 - 4.80T + 7T^{2} \)
11 \( 1 + 4.80T + 11T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 - 5.13T + 19T^{2} \)
23 \( 1 + 3.96T + 23T^{2} \)
29 \( 1 - 1.33T + 29T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 + 3.62T + 37T^{2} \)
41 \( 1 - 7.66T + 41T^{2} \)
43 \( 1 + 2.38T + 43T^{2} \)
47 \( 1 - 5.94T + 47T^{2} \)
53 \( 1 - 0.139T + 53T^{2} \)
59 \( 1 - 5.47T + 59T^{2} \)
61 \( 1 - 4.94T + 61T^{2} \)
67 \( 1 + 6.74T + 67T^{2} \)
71 \( 1 + 1.13T + 71T^{2} \)
73 \( 1 - 5.27T + 73T^{2} \)
79 \( 1 + 4.77T + 79T^{2} \)
83 \( 1 + 1.60T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−8.102098829400259032741422226381, −7.73848525305371960471728038311, −7.21601879929890544657723196023, −5.85574625251574074579957445111, −5.13746139800441031827460045007, −4.46892552812076689336983611688, −3.59264457097128764747637843461, −2.50739878676306923487312739145, −2.19134061192460131251088737841, −1.24631448189815198334064489324, 1.24631448189815198334064489324, 2.19134061192460131251088737841, 2.50739878676306923487312739145, 3.59264457097128764747637843461, 4.46892552812076689336983611688, 5.13746139800441031827460045007, 5.85574625251574074579957445111, 7.21601879929890544657723196023, 7.73848525305371960471728038311, 8.102098829400259032741422226381

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