Properties

Label 2-188760-1.1-c1-0-31
Degree $2$
Conductor $188760$
Sign $-1$
Analytic cond. $1507.25$
Root an. cond. $38.8233$
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  − 3-s − 5-s + 9-s + 13-s + 15-s − 2·17-s + 4·19-s + 25-s − 27-s + 2·29-s − 8·31-s + 10·37-s − 39-s − 10·41-s − 4·43-s − 45-s − 7·49-s + 2·51-s − 14·53-s − 4·57-s + 4·59-s − 6·61-s − 65-s + 8·67-s − 8·71-s − 6·73-s − 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 1.64·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.280·51-s − 1.92·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.124·65-s + 0.977·67-s − 0.949·71-s − 0.702·73-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(188760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1507.25\)
Root analytic conductor: \(38.8233\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{188760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−13.25953118040647, −12.91722276961835, −12.33625243948355, −11.89257400993523, −11.42967820808574, −11.09883546557404, −10.68948466708943, −10.02725340876900, −9.611472610151252, −9.154660150846142, −8.580189526756351, −7.952394056299899, −7.700243636798935, −6.991545612086888, −6.620380218476167, −6.107047529229916, −5.507771092218722, −5.020047260723316, −4.535350623043251, −4.001970211385171, −3.268951297420965, −3.010503727148970, −1.966185517366614, −1.514447015517543, −0.6980236484743491, 0, 0.6980236484743491, 1.514447015517543, 1.966185517366614, 3.010503727148970, 3.268951297420965, 4.001970211385171, 4.535350623043251, 5.020047260723316, 5.507771092218722, 6.107047529229916, 6.620380218476167, 6.991545612086888, 7.700243636798935, 7.952394056299899, 8.580189526756351, 9.154660150846142, 9.611472610151252, 10.02725340876900, 10.68948466708943, 11.09883546557404, 11.42967820808574, 11.89257400993523, 12.33625243948355, 12.91722276961835, 13.25953118040647

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