Properties

Label 2-2475-1.1-c1-0-38
Degree $2$
Conductor $2475$
Sign $-1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 3·7-s + 3·8-s + 11-s + 2·13-s + 3·14-s − 16-s − 3·17-s − 19-s − 22-s − 23-s − 2·26-s + 3·28-s + 6·29-s + 4·31-s − 5·32-s + 3·34-s + 37-s + 38-s − 5·41-s + 4·43-s − 44-s + 46-s − 3·47-s + 2·49-s − 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.13·7-s + 1.06·8-s + 0.301·11-s + 0.554·13-s + 0.801·14-s − 1/4·16-s − 0.727·17-s − 0.229·19-s − 0.213·22-s − 0.208·23-s − 0.392·26-s + 0.566·28-s + 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.514·34-s + 0.164·37-s + 0.162·38-s − 0.780·41-s + 0.609·43-s − 0.150·44-s + 0.147·46-s − 0.437·47-s + 2/7·49-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2475} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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

−8.500775704585080846252634260999, −8.155207854629156628630531146509, −6.88655912806536134012016034996, −6.51780423833610446264208059382, −5.45001030587274355240973905662, −4.42303895506447824218553717818, −3.72455677823649494894558967993, −2.62344002697790795187697994636, −1.23297886210285181777321436748, 0, 1.23297886210285181777321436748, 2.62344002697790795187697994636, 3.72455677823649494894558967993, 4.42303895506447824218553717818, 5.45001030587274355240973905662, 6.51780423833610446264208059382, 6.88655912806536134012016034996, 8.155207854629156628630531146509, 8.500775704585080846252634260999

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