↑

Properties

Label 2-204e2-1.1-c1-0-38
Degree $2$
Conductor $41616$
Sign $-1$
Analytic cond. $332.305$
Root an. cond. $18.2292$
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  − 2·5-s + 4·11-s − 2·13-s − 4·19-s − 25-s − 10·29-s + 8·31-s + 2·37-s + 10·41-s − 12·43-s − 7·49-s − 6·53-s − 8·55-s + 12·59-s + 10·61-s + 4·65-s + 12·67-s − 10·73-s − 8·79-s + 4·83-s + 6·89-s + 8·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s − 0.554·13-s − 0.917·19-s − 1/5·25-s − 1.85·29-s + 1.43·31-s + 0.328·37-s + 1.56·41-s − 1.82·43-s − 49-s − 0.824·53-s − 1.07·55-s + 1.56·59-s + 1.28·61-s + 0.496·65-s + 1.46·67-s − 1.17·73-s − 0.900·79-s + 0.439·83-s + 0.635·89-s + 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41616\)    =    \(2^{4} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(332.305\)
Root analytic conductor: \(18.2292\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 41616,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
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

−14.94086759702950, −14.49220151718706, −14.24401867966012, −13.22419130350640, −12.98636901432716, −12.38056559805778, −11.63640356758508, −11.53586203951884, −11.06348565147513, −10.12363554584897, −9.800482625673156, −9.182683549883779, −8.529728813427527, −8.146649009294826, −7.451611226194109, −7.029562448959072, −6.324243230499370, −5.926772960271698, −4.937191908153281, −4.517515005599864, −3.753717218958132, −3.538488587845638, −2.464190115076650, −1.841868909997775, −0.8786517896260251, 0, 0.8786517896260251, 1.841868909997775, 2.464190115076650, 3.538488587845638, 3.753717218958132, 4.517515005599864, 4.937191908153281, 5.926772960271698, 6.324243230499370, 7.029562448959072, 7.451611226194109, 8.146649009294826, 8.529728813427527, 9.182683549883779, 9.800482625673156, 10.12363554584897, 11.06348565147513, 11.53586203951884, 11.63640356758508, 12.38056559805778, 12.98636901432716, 13.22419130350640, 14.24401867966012, 14.49220151718706, 14.94086759702950

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