Properties

Label 2-5712-1.1-c1-0-40
Degree $2$
Conductor $5712$
Sign $1$
Analytic cond. $45.6105$
Root an. cond. $6.75355$
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 + 3·5-s + 7-s + 9-s − 11-s + 13-s + 3·15-s − 17-s − 6·19-s + 21-s + 2·23-s + 4·25-s + 27-s − 2·29-s − 33-s + 3·35-s + 5·37-s + 39-s + 4·41-s + 9·43-s + 3·45-s + 49-s − 51-s + 11·53-s − 3·55-s − 6·57-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.774·15-s − 0.242·17-s − 1.37·19-s + 0.218·21-s + 0.417·23-s + 4/5·25-s + 0.192·27-s − 0.371·29-s − 0.174·33-s + 0.507·35-s + 0.821·37-s + 0.160·39-s + 0.624·41-s + 1.37·43-s + 0.447·45-s + 1/7·49-s − 0.140·51-s + 1.51·53-s − 0.404·55-s − 0.794·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5712\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(45.6105\)
Root analytic conductor: \(6.75355\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5712} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5712,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.476984177\)
\(L(\frac12)\) \(\approx\) \(3.476984177\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−8.230181233163377072391206556685, −7.46789639665146044636445370509, −6.61533913687974552980184755280, −6.01170978697682863588758156995, −5.30740827986631055992699318293, −4.46932263830009852693651038810, −3.66833515921551500358074348792, −2.37599584843001900977226187836, −2.21361673138795484971928240020, −0.994417332982276826321481545703, 0.994417332982276826321481545703, 2.21361673138795484971928240020, 2.37599584843001900977226187836, 3.66833515921551500358074348792, 4.46932263830009852693651038810, 5.30740827986631055992699318293, 6.01170978697682863588758156995, 6.61533913687974552980184755280, 7.46789639665146044636445370509, 8.230181233163377072391206556685

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