Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 7-s + 9-s + 11-s + 6·13-s − 2·15-s + 2·17-s − 4·19-s − 21-s − 25-s + 27-s − 2·29-s − 8·31-s + 33-s + 2·35-s + 6·37-s + 6·39-s + 10·41-s + 4·43-s − 2·45-s + 8·47-s + 49-s + 2·51-s + 6·53-s − 2·55-s − 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.174·33-s + 0.338·35-s + 0.986·37-s + 0.960·39-s + 1.56·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.269·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3696} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3696,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.976846717$
$L(\frac12)$  $\approx$  $1.976846717$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.56536356410111, −17.74066245626067, −16.83105662773242, −16.30837328754032, −15.71360495815739, −15.26594172259500, −14.51314426100873, −13.98305048874967, −13.12017990469149, −12.74683464667079, −11.97524385482091, −11.08792443828321, −10.83914866198611, −9.811668872988357, −9.006853114794629, −8.626329770389122, −7.738148101392036, −7.312170845961964, −6.230157594706294, −5.747994928802587, −4.312265473336237, −3.888452420429258, −3.204746073160933, −2.039960551080887, −0.8094509313004489, 0.8094509313004489, 2.039960551080887, 3.204746073160933, 3.888452420429258, 4.312265473336237, 5.747994928802587, 6.230157594706294, 7.312170845961964, 7.738148101392036, 8.626329770389122, 9.006853114794629, 9.811668872988357, 10.83914866198611, 11.08792443828321, 11.97524385482091, 12.74683464667079, 13.12017990469149, 13.98305048874967, 14.51314426100873, 15.26594172259500, 15.71360495815739, 16.30837328754032, 16.83105662773242, 17.74066245626067, 18.56536356410111

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