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Properties

Label 2-5408-1.1-c1-0-80
Degree $2$
Conductor $5408$
Sign $-1$
Analytic cond. $43.1830$
Root an. cond. $6.57138$
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  − 4·5-s − 3·9-s + 2·17-s + 11·25-s + 10·29-s + 12·37-s − 8·41-s + 12·45-s − 7·49-s + 14·53-s − 10·61-s − 16·73-s + 9·81-s − 8·85-s − 16·89-s − 8·97-s + 2·101-s − 20·109-s + 14·113-s + ⋯
L(s)  = 1  − 1.78·5-s − 9-s + 0.485·17-s + 11/5·25-s + 1.85·29-s + 1.97·37-s − 1.24·41-s + 1.78·45-s − 49-s + 1.92·53-s − 1.28·61-s − 1.87·73-s + 81-s − 0.867·85-s − 1.69·89-s − 0.812·97-s + 0.199·101-s − 1.91·109-s + 1.31·113-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5408,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.988612977350406093974206442013, −7.21424279733010872572454801516, −6.49364081247895273094607913476, −5.62302150895633800615973468947, −4.70807460073308666993587405749, −4.14113725726325628790675953937, −3.21139015568415693943466713575, −2.72602212485500455177659066469, −1.04864477909657256988988373490, 0, 1.04864477909657256988988373490, 2.72602212485500455177659066469, 3.21139015568415693943466713575, 4.14113725726325628790675953937, 4.70807460073308666993587405749, 5.62302150895633800615973468947, 6.49364081247895273094607913476, 7.21424279733010872572454801516, 7.988612977350406093974206442013

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