Properties

Label 2-29645-1.1-c1-0-0
Degree $2$
Conductor $29645$
Sign $1$
Analytic cond. $236.716$
Root an. cond. $15.3855$
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 − 4-s − 5-s − 3·8-s − 3·9-s − 10-s + 4·13-s − 16-s − 3·18-s − 4·19-s + 20-s − 6·23-s + 25-s + 4·26-s − 6·31-s + 5·32-s + 3·36-s + 2·37-s − 4·38-s + 3·40-s + 2·41-s + 12·43-s + 3·45-s − 6·46-s − 8·47-s + 50-s − 4·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 9-s − 0.316·10-s + 1.10·13-s − 1/4·16-s − 0.707·18-s − 0.917·19-s + 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.784·26-s − 1.07·31-s + 0.883·32-s + 1/2·36-s + 0.328·37-s − 0.648·38-s + 0.474·40-s + 0.312·41-s + 1.82·43-s + 0.447·45-s − 0.884·46-s − 1.16·47-s + 0.141·50-s − 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29645\)    =    \(5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(236.716\)
Root analytic conductor: \(15.3855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{29645} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29645,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7639083304\)
\(L(\frac12)\) \(\approx\) \(0.7639083304\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.05027221443307, −14.45229957696014, −14.12040762636433, −13.68928427211074, −12.96653901603431, −12.63052458278843, −12.04833594633228, −11.51336300046942, −10.87708671644025, −10.61110735598687, −9.537616789840064, −9.131779185725032, −8.666847207933157, −7.949161190998839, −7.732617476572994, −6.431190442356906, −6.197082166203930, −5.678043522329501, −4.918666560343386, −4.297797439963999, −3.762997452416668, −3.224013189581558, −2.495617796766924, −1.502405300805963, −0.2989023889619969, 0.2989023889619969, 1.502405300805963, 2.495617796766924, 3.224013189581558, 3.762997452416668, 4.297797439963999, 4.918666560343386, 5.678043522329501, 6.197082166203930, 6.431190442356906, 7.732617476572994, 7.949161190998839, 8.666847207933157, 9.131779185725032, 9.537616789840064, 10.61110735598687, 10.87708671644025, 11.51336300046942, 12.04833594633228, 12.63052458278843, 12.96653901603431, 13.68928427211074, 14.12040762636433, 14.45229957696014, 15.05027221443307

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