Properties

Label 2-13552-1.1-c1-0-4
Degree $2$
Conductor $13552$
Sign $1$
Analytic cond. $108.213$
Root an. cond. $10.4025$
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  + 2·5-s − 7-s − 3·9-s − 2·13-s + 6·17-s + 8·19-s − 25-s − 6·29-s − 8·31-s − 2·35-s − 2·37-s − 2·41-s − 4·43-s − 6·45-s + 8·47-s + 49-s + 6·53-s + 6·61-s + 3·63-s − 4·65-s + 4·67-s + 8·71-s − 10·73-s + 16·79-s + 9·81-s + 8·83-s + 12·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 9-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.894·45-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.768·61-s + 0.377·63-s − 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 1.80·79-s + 81-s + 0.878·83-s + 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13552\)    =    \(2^{4} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(108.213\)
Root analytic conductor: \(10.4025\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{13552} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 13552,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.109489904\)
\(L(\frac12)\) \(\approx\) \(2.109489904\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−16.55396780737299, −15.65107129434973, −14.93465069367051, −14.43804506389249, −13.93938499767119, −13.56207845299293, −12.83742592947578, −12.18105184557004, −11.74279248999077, −11.12047980717424, −10.32184855885086, −9.813761635026272, −9.360540366280823, −8.872019738763444, −7.882546856710184, −7.484410531599589, −6.768784368125564, −5.836485557210028, −5.449971086197904, −5.183783273263211, −3.748711321592123, −3.315325009962182, −2.501208561951002, −1.705643996975220, −0.6412279080455006, 0.6412279080455006, 1.705643996975220, 2.501208561951002, 3.315325009962182, 3.748711321592123, 5.183783273263211, 5.449971086197904, 5.836485557210028, 6.768784368125564, 7.484410531599589, 7.882546856710184, 8.872019738763444, 9.360540366280823, 9.813761635026272, 10.32184855885086, 11.12047980717424, 11.74279248999077, 12.18105184557004, 12.83742592947578, 13.56207845299293, 13.93938499767119, 14.43804506389249, 14.93465069367051, 15.65107129434973, 16.55396780737299

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