Properties

Degree $2$
Conductor $336$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 18·5-s − 7·7-s + 9·9-s + 36·11-s − 34·13-s − 54·15-s + 42·17-s + 124·19-s − 21·21-s + 199·25-s + 27·27-s + 102·29-s + 160·31-s + 108·33-s + 126·35-s + 398·37-s − 102·39-s − 318·41-s + 268·43-s − 162·45-s − 240·47-s + 49·49-s + 126·51-s − 498·53-s − 648·55-s + 372·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.60·5-s − 0.377·7-s + 1/3·9-s + 0.986·11-s − 0.725·13-s − 0.929·15-s + 0.599·17-s + 1.49·19-s − 0.218·21-s + 1.59·25-s + 0.192·27-s + 0.653·29-s + 0.926·31-s + 0.569·33-s + 0.608·35-s + 1.76·37-s − 0.418·39-s − 1.21·41-s + 0.950·43-s − 0.536·45-s − 0.744·47-s + 1/7·49-s + 0.345·51-s − 1.29·53-s − 1.58·55-s + 0.864·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{336} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.676142833\)
\(L(\frac12)\) \(\approx\) \(1.676142833\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
7 \( 1 + p T \)
good5 \( 1 + 18 T + p^{3} T^{2} \)
11 \( 1 - 36 T + p^{3} T^{2} \)
13 \( 1 + 34 T + p^{3} T^{2} \)
17 \( 1 - 42 T + p^{3} T^{2} \)
19 \( 1 - 124 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 - 102 T + p^{3} T^{2} \)
31 \( 1 - 160 T + p^{3} T^{2} \)
37 \( 1 - 398 T + p^{3} T^{2} \)
41 \( 1 + 318 T + p^{3} T^{2} \)
43 \( 1 - 268 T + p^{3} T^{2} \)
47 \( 1 + 240 T + p^{3} T^{2} \)
53 \( 1 + 498 T + p^{3} T^{2} \)
59 \( 1 - 132 T + p^{3} T^{2} \)
61 \( 1 - 398 T + p^{3} T^{2} \)
67 \( 1 + 92 T + p^{3} T^{2} \)
71 \( 1 - 720 T + p^{3} T^{2} \)
73 \( 1 + 502 T + p^{3} T^{2} \)
79 \( 1 - 1024 T + p^{3} T^{2} \)
83 \( 1 - 204 T + p^{3} T^{2} \)
89 \( 1 - 354 T + p^{3} T^{2} \)
97 \( 1 + 286 T + p^{3} T^{2} \)
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

−11.43808645846862009442108850231, −10.02564470085939608724930382948, −9.253989239769519311151869418084, −8.108753058559954913654317784987, −7.52899283861443377884794982206, −6.54325801361683782115600262154, −4.84914072840988045227180385771, −3.81155493369603893403057077384, −2.96531272172107702079338172136, −0.877631644606296641782168486277, 0.877631644606296641782168486277, 2.96531272172107702079338172136, 3.81155493369603893403057077384, 4.84914072840988045227180385771, 6.54325801361683782115600262154, 7.52899283861443377884794982206, 8.108753058559954913654317784987, 9.253989239769519311151869418084, 10.02564470085939608724930382948, 11.43808645846862009442108850231

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