Properties

Degree $2$
Conductor $25857$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s + 4·11-s − 16-s − 17-s + 4·19-s + 2·20-s − 4·22-s − 25-s + 2·29-s + 8·31-s − 5·32-s + 34-s + 2·37-s − 4·38-s − 6·40-s + 2·41-s − 4·43-s − 4·44-s + 8·47-s − 7·49-s + 50-s + 10·53-s − 8·55-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s + 1.20·11-s − 1/4·16-s − 0.242·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.883·32-s + 0.171·34-s + 0.328·37-s − 0.648·38-s − 0.948·40-s + 0.312·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s − 49-s + 0.141·50-s + 1.37·53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25857 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25857 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25857\)    =    \(3^{2} \cdot 13^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{25857} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25857,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.269275476\)
\(L(\frac12)\) \(\approx\) \(1.269275476\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
17 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−15.44738411850052, −14.80973160836216, −14.17165419368992, −13.85303698305337, −13.23019832227789, −12.56429288800168, −11.93350472053664, −11.52864011764867, −11.10212633200585, −10.18039927686252, −9.860480216193630, −9.283920992614683, −8.642642932281908, −8.266958214271655, −7.703151164637540, −7.046458296265620, −6.591811528864091, −5.675506981928342, −4.985227897120660, −4.215608609077222, −3.946098759537343, −3.156390107361832, −2.137004725827043, −1.108590101043256, −0.6232056283256693, 0.6232056283256693, 1.108590101043256, 2.137004725827043, 3.156390107361832, 3.946098759537343, 4.215608609077222, 4.985227897120660, 5.675506981928342, 6.591811528864091, 7.046458296265620, 7.703151164637540, 8.266958214271655, 8.642642932281908, 9.283920992614683, 9.860480216193630, 10.18039927686252, 11.10212633200585, 11.52864011764867, 11.93350472053664, 12.56429288800168, 13.23019832227789, 13.85303698305337, 14.17165419368992, 14.80973160836216, 15.44738411850052

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