↑

Properties

Label 2-392-8.3-c0-0-2
Degree $2$
Conductor $392$
Sign $1$
Analytic cond. $0.195633$
Root an. cond. $0.442304$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 9-s − 2·11-s + 16-s − 18-s − 2·22-s + 25-s + 32-s − 36-s − 2·43-s − 2·44-s + 50-s + 64-s + 2·67-s − 72-s + 81-s − 2·86-s − 2·88-s + 2·99-s + 100-s + 2·107-s − 2·113-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s − 9-s − 2·11-s + 16-s − 18-s − 2·22-s + 25-s + 32-s − 36-s − 2·43-s − 2·44-s + 50-s + 64-s + 2·67-s − 72-s + 81-s − 2·86-s − 2·88-s + 2·99-s + 100-s + 2·107-s − 2·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.195633\)
Root analytic conductor: \(0.442304\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{392} (99, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.326103946\)
\(L(\frac12)\) \(\approx\) \(1.326103946\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 \)
good3 \( 1 + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 + T )^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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

−11.55457175682676341145374708872, −10.82302225200663020125117006648, −10.06140452427577823324442250013, −8.502992115377592723768747076372, −7.74588668058234013467269768806, −6.61326218356275638677595046831, −5.48005157786240775431542969066, −4.91053368894548166897594584151, −3.30538355832891161813344614001, −2.41597639036909237292532362178, 2.41597639036909237292532362178, 3.30538355832891161813344614001, 4.91053368894548166897594584151, 5.48005157786240775431542969066, 6.61326218356275638677595046831, 7.74588668058234013467269768806, 8.502992115377592723768747076372, 10.06140452427577823324442250013, 10.82302225200663020125117006648, 11.55457175682676341145374708872

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