Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{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  − 2·7-s + 11-s − 4·13-s − 2·17-s + 23-s − 7·31-s − 3·37-s + 8·41-s − 6·43-s − 8·47-s − 3·49-s − 6·53-s + 5·59-s + 12·61-s − 7·67-s − 3·71-s − 4·73-s − 2·77-s + 10·79-s + 6·83-s − 15·89-s + 8·91-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.301·11-s − 1.10·13-s − 0.485·17-s + 0.208·23-s − 1.25·31-s − 0.493·37-s + 1.24·41-s − 0.914·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.650·59-s + 1.53·61-s − 0.855·67-s − 0.356·71-s − 0.468·73-s − 0.227·77-s + 1.12·79-s + 0.658·83-s − 1.58·89-s + 0.838·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(39600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{39600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 39600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8192710830$
$L(\frac12)$  $\approx$  $0.8192710830$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.75926281733095, −14.41111458083851, −13.73843017448697, −13.10378771850520, −12.76894349467224, −12.30790372749137, −11.62293216446684, −11.21026181058708, −10.54756065599058, −9.966787628479209, −9.485833426162263, −9.123988494128151, −8.399321207327632, −7.799299595540175, −7.117986170101908, −6.757004810682732, −6.165305201768287, −5.413257902950058, −4.924042694674779, −4.198468048056038, −3.551345965049679, −2.918238330464930, −2.229013145737167, −1.473965456016027, −0.3226584835751798, 0.3226584835751798, 1.473965456016027, 2.229013145737167, 2.918238330464930, 3.551345965049679, 4.198468048056038, 4.924042694674779, 5.413257902950058, 6.165305201768287, 6.757004810682732, 7.117986170101908, 7.799299595540175, 8.399321207327632, 9.123988494128151, 9.485833426162263, 9.966787628479209, 10.54756065599058, 11.21026181058708, 11.62293216446684, 12.30790372749137, 12.76894349467224, 13.10378771850520, 13.73843017448697, 14.41111458083851, 14.75926281733095

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