Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} \cdot 13 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 13-s + 17-s − 2·31-s + 35-s − 37-s + 43-s − 47-s + 65-s − 71-s + 85-s + 91-s + 2·107-s − 109-s − 2·113-s + 119-s + ⋯
L(s)  = 1  + 5-s + 7-s + 13-s + 17-s − 2·31-s + 35-s − 37-s + 43-s − 47-s + 65-s − 71-s + 85-s + 91-s + 2·107-s − 109-s − 2·113-s + 119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{3744} (2287, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3744,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.832426030\)
\(L(\frac12)\)  \(\approx\)  \(1.832426030\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;13\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 - T \)
good5 \( 1 - T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( 1 - T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 + T )^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.743482093758118961320489667409, −7.967386679435940305650168263944, −7.30898559345027008254216268681, −6.33190715670784493096816996215, −5.61260268309217016152046500407, −5.16702174510769087330189145955, −4.07034202281980020039752522804, −3.21753954892597903232864497630, −1.96969571707703874645104744886, −1.38198762194799550133640712557, 1.38198762194799550133640712557, 1.96969571707703874645104744886, 3.21753954892597903232864497630, 4.07034202281980020039752522804, 5.16702174510769087330189145955, 5.61260268309217016152046500407, 6.33190715670784493096816996215, 7.30898559345027008254216268681, 7.967386679435940305650168263944, 8.743482093758118961320489667409

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