Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17 $
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 − 4·7-s + 9-s + 4·11-s + 13-s + 17-s + 8·19-s + 4·21-s − 8·23-s − 27-s − 10·29-s + 8·31-s − 4·33-s + 6·37-s − 39-s + 2·41-s + 4·43-s + 8·47-s + 9·49-s − 51-s − 6·53-s − 8·57-s + 4·59-s − 14·61-s − 4·63-s + 4·67-s + 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.242·17-s + 1.83·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.160·39-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.140·51-s − 0.824·53-s − 1.05·57-s + 0.520·59-s − 1.79·61-s − 0.503·63-s + 0.488·67-s + 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(265200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{265200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 265200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.734843742$
$L(\frac12)$  $\approx$  $1.734843742$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.63041718659697, −12.31035357856820, −11.95899400675650, −11.41882908449147, −11.13848638549321, −10.40703178969384, −9.837983813637779, −9.630886860313187, −9.343753650581853, −8.765194178238775, −7.995381809571493, −7.533534045038678, −7.139603415882333, −6.520334761991023, −6.082236839088096, −5.848312727260233, −5.353038673568339, −4.438623521243382, −4.076648577427238, −3.542921978355004, −3.128004300651214, −2.437928323899957, −1.638817824785697, −0.9990605302070040, −0.4381322650701255, 0.4381322650701255, 0.9990605302070040, 1.638817824785697, 2.437928323899957, 3.128004300651214, 3.542921978355004, 4.076648577427238, 4.438623521243382, 5.353038673568339, 5.848312727260233, 6.082236839088096, 6.520334761991023, 7.139603415882333, 7.533534045038678, 7.995381809571493, 8.765194178238775, 9.343753650581853, 9.630886860313187, 9.837983813637779, 10.40703178969384, 11.13848638549321, 11.41882908449147, 11.95899400675650, 12.31035357856820, 12.63041718659697

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