↑

Properties

Label 2-39600-1.1-c1-0-120
Degree $2$
Conductor $39600$
Sign $-1$
Analytic cond. $316.207$
Root an. cond. $17.7822$
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  + 2·7-s + 11-s − 2·13-s + 6·17-s − 2·19-s − 6·29-s + 4·31-s − 2·37-s + 6·41-s − 10·43-s + 12·47-s − 3·49-s + 12·53-s − 12·59-s − 10·61-s + 8·67-s − 12·71-s − 14·73-s + 2·77-s − 2·79-s − 12·83-s − 4·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.755·7-s + 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.458·19-s − 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s − 1.52·43-s + 1.75·47-s − 3/7·49-s + 1.64·53-s − 1.56·59-s − 1.28·61-s + 0.977·67-s − 1.42·71-s − 1.63·73-s + 0.227·77-s − 0.225·79-s − 1.31·83-s − 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(316.207\)
Root analytic conductor: \(17.7822\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 39600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−15.02705204746111, −14.51322495597745, −14.14346354387483, −13.57144408070731, −12.96126788557094, −12.32187733772076, −11.97734666793843, −11.46393814157204, −10.89171279809067, −10.19367724392999, −9.973514697249469, −9.108535423449352, −8.747683272886998, −8.024302470216002, −7.513587458487541, −7.183336561586019, −6.271832177432934, −5.735173095207108, −5.212471020350174, −4.521267384987907, −4.002839591007268, −3.215423536932673, −2.560201901155216, −1.699038730297384, −1.134366352490380, 0, 1.134366352490380, 1.699038730297384, 2.560201901155216, 3.215423536932673, 4.002839591007268, 4.521267384987907, 5.212471020350174, 5.735173095207108, 6.271832177432934, 7.183336561586019, 7.513587458487541, 8.024302470216002, 8.747683272886998, 9.108535423449352, 9.973514697249469, 10.19367724392999, 10.89171279809067, 11.46393814157204, 11.97734666793843, 12.32187733772076, 12.96126788557094, 13.57144408070731, 14.14346354387483, 14.51322495597745, 15.02705204746111

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