Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 2·13-s − 6·17-s + 4·19-s − 6·23-s − 6·29-s + 4·31-s − 2·37-s − 6·41-s − 10·43-s + 6·47-s − 3·49-s − 6·53-s + 12·59-s + 2·61-s + 2·67-s − 12·71-s − 2·73-s − 8·79-s − 6·83-s + 6·89-s − 4·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.755·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s − 1.52·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.244·67-s − 1.42·71-s − 0.234·73-s − 0.900·79-s − 0.658·83-s + 0.635·89-s − 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{3600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 3600,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−18.46820140479374, −17.97559040763724, −17.42819666903226, −16.89623114956044, −15.98477057426138, −15.59968374704994, −14.79707183137033, −14.33446731345539, −13.53150641577994, −13.13794256988798, −12.10486186709443, −11.64416341351227, −11.11015959875968, −10.19730125648889, −9.701233901505577, −8.775680456749499, −8.239614848469988, −7.440589718159377, −6.802591952809436, −5.890441824313173, −5.066224028090545, −4.436815667009123, −3.514232515812804, −2.375748260077808, −1.589202744511732, 0, 1.589202744511732, 2.375748260077808, 3.514232515812804, 4.436815667009123, 5.066224028090545, 5.890441824313173, 6.802591952809436, 7.440589718159377, 8.239614848469988, 8.775680456749499, 9.701233901505577, 10.19730125648889, 11.11015959875968, 11.64416341351227, 12.10486186709443, 13.13794256988798, 13.53150641577994, 14.33446731345539, 14.79707183137033, 15.59968374704994, 15.98477057426138, 16.89623114956044, 17.42819666903226, 17.97559040763724, 18.46820140479374

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