↑

Properties

Label 2-210e2-1.1-c1-0-61
Degree $2$
Conductor $44100$
Sign $-1$
Analytic cond. $352.140$
Root an. cond. $18.7654$
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  − 11-s − 4·13-s + 2·17-s + 4·19-s − 7·23-s + 9·29-s + 2·31-s + 37-s + 8·41-s − 9·43-s + 4·47-s − 6·53-s + 4·59-s − 4·61-s + 9·67-s − 5·71-s − 10·73-s − 15·79-s − 6·83-s + 8·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.301·11-s − 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.45·23-s + 1.67·29-s + 0.359·31-s + 0.164·37-s + 1.24·41-s − 1.37·43-s + 0.583·47-s − 0.824·53-s + 0.520·59-s − 0.512·61-s + 1.09·67-s − 0.593·71-s − 1.17·73-s − 1.68·79-s − 0.658·83-s + 0.847·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(352.140\)
Root analytic conductor: \(18.7654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 44100,\ (\ :1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + T + p T^{2} \) 1.11.b
13 \( 1 + 4 T + p T^{2} \) 1.13.e
17 \( 1 - 2 T + p T^{2} \) 1.17.ac
19 \( 1 - 4 T + p T^{2} \) 1.19.ae
23 \( 1 + 7 T + p T^{2} \) 1.23.h
29 \( 1 - 9 T + p T^{2} \) 1.29.aj
31 \( 1 - 2 T + p T^{2} \) 1.31.ac
37 \( 1 - T + p T^{2} \) 1.37.ab
41 \( 1 - 8 T + p T^{2} \) 1.41.ai
43 \( 1 + 9 T + p T^{2} \) 1.43.j
47 \( 1 - 4 T + p T^{2} \) 1.47.ae
53 \( 1 + 6 T + p T^{2} \) 1.53.g
59 \( 1 - 4 T + p T^{2} \) 1.59.ae
61 \( 1 + 4 T + p T^{2} \) 1.61.e
67 \( 1 - 9 T + p T^{2} \) 1.67.aj
71 \( 1 + 5 T + p T^{2} \) 1.71.f
73 \( 1 + 10 T + p T^{2} \) 1.73.k
79 \( 1 + 15 T + p T^{2} \) 1.79.p
83 \( 1 + 6 T + p T^{2} \) 1.83.g
89 \( 1 - 8 T + p T^{2} \) 1.89.ai
97 \( 1 + 10 T + p T^{2} \) 1.97.k
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

−14.77958580169016, −14.37170208359099, −14.00657768128940, −13.43672689960468, −12.81361571396025, −12.21860519430069, −11.92239963742930, −11.46160414496617, −10.61877913575237, −10.14695015240517, −9.775929198737916, −9.307046601585891, −8.358781917252210, −8.145613381557215, −7.407591260806258, −7.049678525501748, −6.212711926069787, −5.764984430925019, −5.058798608233593, −4.567950847482659, −3.930866772778692, −3.022371430126659, −2.659825485255379, −1.800063582056930, −0.9471331106443235, 0, 0.9471331106443235, 1.800063582056930, 2.659825485255379, 3.022371430126659, 3.930866772778692, 4.567950847482659, 5.058798608233593, 5.764984430925019, 6.212711926069787, 7.049678525501748, 7.407591260806258, 8.145613381557215, 8.358781917252210, 9.307046601585891, 9.775929198737916, 10.14695015240517, 10.61877913575237, 11.46160414496617, 11.92239963742930, 12.21860519430069, 12.81361571396025, 13.43672689960468, 14.00657768128940, 14.37170208359099, 14.77958580169016

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