↑

Properties

Label 2-125235-1.1-c1-0-10
Degree $2$
Conductor $125235$
Sign $1$
Analytic cond. $1000.00$
Root an. cond. $31.6228$
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  − 2·2-s + 2·4-s − 5-s + 5·7-s + 2·10-s + 6·13-s − 10·14-s − 4·16-s + 17-s − 2·19-s − 2·20-s + 23-s + 25-s − 12·26-s + 10·28-s − 29-s − 5·31-s + 8·32-s − 2·34-s − 5·35-s − 7·37-s + 4·38-s − 7·41-s + 8·43-s − 2·46-s + 12·47-s + 18·49-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s + 1.88·7-s + 0.632·10-s + 1.66·13-s − 2.67·14-s − 16-s + 0.242·17-s − 0.458·19-s − 0.447·20-s + 0.208·23-s + 1/5·25-s − 2.35·26-s + 1.88·28-s − 0.185·29-s − 0.898·31-s + 1.41·32-s − 0.342·34-s − 0.845·35-s − 1.15·37-s + 0.648·38-s − 1.09·41-s + 1.21·43-s − 0.294·46-s + 1.75·47-s + 18/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125235\)    =    \(3^{2} \cdot 5 \cdot 11^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1000.00\)
Root analytic conductor: \(31.6228\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 125235,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.457699186\)
\(L(\frac12)\) \(\approx\) \(1.457699186\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 \)
23 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−13.64831417065205, −13.07917793191342, −12.28404815621864, −11.91866785254065, −11.31499348582074, −10.92345172188417, −10.67026529641090, −10.38949077445666, −9.332567612910159, −9.030678610479995, −8.597213104919299, −8.224496090212030, −7.744159078693810, −7.412824760376150, −6.813291629321172, −6.104138930507955, −5.507666427698231, −4.914064853993240, −4.305800792999217, −3.881664880253109, −3.101911160170513, −2.147821977616067, −1.620341621837184, −1.255588804515115, −0.5060854533798274, 0.5060854533798274, 1.255588804515115, 1.620341621837184, 2.147821977616067, 3.101911160170513, 3.881664880253109, 4.305800792999217, 4.914064853993240, 5.507666427698231, 6.104138930507955, 6.813291629321172, 7.412824760376150, 7.744159078693810, 8.224496090212030, 8.597213104919299, 9.030678610479995, 9.332567612910159, 10.38949077445666, 10.67026529641090, 10.92345172188417, 11.31499348582074, 11.91866785254065, 12.28404815621864, 13.07917793191342, 13.64831417065205

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