Properties

Label 2-38808-1.1-c1-0-29
Degree $2$
Conductor $38808$
Sign $1$
Analytic cond. $309.883$
Root an. cond. $17.6035$
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  + 2·5-s − 11-s + 6·13-s − 2·17-s + 8·19-s + 4·23-s − 25-s − 2·29-s + 8·31-s + 6·37-s − 2·41-s + 8·43-s − 4·47-s − 2·53-s − 2·55-s − 12·59-s − 10·61-s + 12·65-s − 12·67-s + 12·71-s − 10·73-s − 8·79-s + 12·83-s − 4·85-s + 10·89-s + 16·95-s + 14·97-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.301·11-s + 1.66·13-s − 0.485·17-s + 1.83·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.986·37-s − 0.312·41-s + 1.21·43-s − 0.583·47-s − 0.274·53-s − 0.269·55-s − 1.56·59-s − 1.28·61-s + 1.48·65-s − 1.46·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s + 1.31·83-s − 0.433·85-s + 1.05·89-s + 1.64·95-s + 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38808\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(309.883\)
Root analytic conductor: \(17.6035\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{38808} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38808,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.763877665\)
\(L(\frac12)\) \(\approx\) \(3.763877665\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.81135093677097, −14.07271628374106, −13.72074544773978, −13.35742994267789, −13.00232349496685, −12.14321610325124, −11.67757133452533, −11.02494550814592, −10.73181417426622, −9.996341476689533, −9.477238125587071, −9.084854484056360, −8.492012069666120, −7.769794058829883, −7.368537350441861, −6.482585722846666, −6.009735059985510, −5.715123948442000, −4.832302234238119, −4.396115457498189, −3.302200304605249, −3.092362408651668, −2.119505565500325, −1.366092473434093, −0.7791185295496685, 0.7791185295496685, 1.366092473434093, 2.119505565500325, 3.092362408651668, 3.302200304605249, 4.396115457498189, 4.832302234238119, 5.715123948442000, 6.009735059985510, 6.482585722846666, 7.368537350441861, 7.769794058829883, 8.492012069666120, 9.084854484056360, 9.477238125587071, 9.996341476689533, 10.73181417426622, 11.02494550814592, 11.67757133452533, 12.14321610325124, 13.00232349496685, 13.35742994267789, 13.72074544773978, 14.07271628374106, 14.81135093677097

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