Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7^{2} \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  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s + 4·13-s + 16-s + 6·17-s − 18-s + 4·19-s + 22-s + 6·23-s + 24-s − 5·25-s − 4·26-s − 27-s + 6·29-s − 8·31-s − 32-s + 33-s − 6·34-s + 36-s − 10·37-s − 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s − 1.64·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3234} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3234,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.181910159$
$L(\frac12)$  $\approx$  $1.181910159$
$L(\frac{3}{2})$   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 \( 1 + T \)
3 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−18.55205984762574, −17.78975707758709, −17.40760249204726, −16.54230572746774, −16.21454743608808, −15.55019791570841, −14.96251848257759, −13.91598015741483, −13.56099984063983, −12.38470240486841, −12.18055671436835, −11.23995414963296, −10.73556731815374, −10.14591835463567, −9.355111614545300, −8.731622022691465, −7.826455456985472, −7.311569382262391, −6.462850197032430, −5.621221867702407, −5.141101816886079, −3.771744023084131, −3.083751570423711, −1.671698232918294, −0.7929031164761629, 0.7929031164761629, 1.671698232918294, 3.083751570423711, 3.771744023084131, 5.141101816886079, 5.621221867702407, 6.462850197032430, 7.311569382262391, 7.826455456985472, 8.731622022691465, 9.355111614545300, 10.14591835463567, 10.73556731815374, 11.23995414963296, 12.18055671436835, 12.38470240486841, 13.56099984063983, 13.91598015741483, 14.96251848257759, 15.55019791570841, 16.21454743608808, 16.54230572746774, 17.40760249204726, 17.78975707758709, 18.55205984762574

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