Properties

Label 2-236992-1.1-c1-0-26
Degree $2$
Conductor $236992$
Sign $1$
Analytic cond. $1892.39$
Root an. cond. $43.5016$
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  + 2·3-s − 7-s + 9-s + 6·17-s + 6·19-s − 2·21-s − 5·25-s − 4·27-s + 6·29-s + 8·31-s + 2·37-s − 2·41-s − 8·43-s − 8·47-s + 49-s + 12·51-s − 2·53-s + 12·57-s + 6·59-s − 63-s − 12·67-s − 8·71-s − 6·73-s − 10·75-s + 16·79-s − 11·81-s + 6·83-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.45·17-s + 1.37·19-s − 0.436·21-s − 25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 1.68·51-s − 0.274·53-s + 1.58·57-s + 0.781·59-s − 0.125·63-s − 1.46·67-s − 0.949·71-s − 0.702·73-s − 1.15·75-s + 1.80·79-s − 1.22·81-s + 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(236992\)    =    \(2^{6} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1892.39\)
Root analytic conductor: \(43.5016\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{236992} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.258187311\)
\(L(\frac12)\) \(\approx\) \(4.258187311\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
23 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−13.24785251925223, −12.32049733038955, −11.97004971174728, −11.71161024956699, −11.13186148732095, −10.24371093081516, −9.942187548021264, −9.787572939851736, −9.182734885545141, −8.545812298058480, −8.298332120963735, −7.655873111419865, −7.523439693292140, −6.807313956329382, −6.072034536859189, −5.912700654239842, −4.969705474315885, −4.802665458525208, −3.814123525336070, −3.471454625192857, −3.001839281110566, −2.653649680565848, −1.818108578211511, −1.259038105022751, −0.5364673163082127, 0.5364673163082127, 1.259038105022751, 1.818108578211511, 2.653649680565848, 3.001839281110566, 3.471454625192857, 3.814123525336070, 4.802665458525208, 4.969705474315885, 5.912700654239842, 6.072034536859189, 6.807313956329382, 7.523439693292140, 7.655873111419865, 8.298332120963735, 8.545812298058480, 9.182734885545141, 9.787572939851736, 9.942187548021264, 10.24371093081516, 11.13186148732095, 11.71161024956699, 11.97004971174728, 12.32049733038955, 13.24785251925223

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