Properties

Label 2-5610-1.1-c1-0-37
Degree $2$
Conductor $5610$
Sign $1$
Analytic cond. $44.7960$
Root an. cond. $6.69298$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 4·13-s − 2·14-s + 15-s + 16-s + 17-s − 18-s + 8·19-s + 20-s + 2·21-s + 22-s − 24-s + 25-s + 4·26-s + 27-s + 2·28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.436·21-s + 0.213·22-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5610\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(44.7960\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.219726501\)
\(L(\frac12)\) \(\approx\) \(2.219726501\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
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 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.014577732573335157216093461931, −7.72353265260683119088373234049, −6.93246209178277148027600484537, −6.15247408446868593637176285741, −5.04880624267662423450314672098, −4.76995000226718477111472880364, −3.31819312970869046252227802193, −2.70729266496666967945316346561, −1.81334445828822228623030230293, −0.894276120626598724062637586377, 0.894276120626598724062637586377, 1.81334445828822228623030230293, 2.70729266496666967945316346561, 3.31819312970869046252227802193, 4.76995000226718477111472880364, 5.04880624267662423450314672098, 6.15247408446868593637176285741, 6.93246209178277148027600484537, 7.72353265260683119088373234049, 8.014577732573335157216093461931

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