Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·7-s + 9-s − 2·13-s − 4·19-s − 4·21-s + 6·23-s + 4·27-s − 6·29-s + 4·31-s + 2·37-s + 4·39-s − 6·41-s + 10·43-s + 6·47-s − 3·49-s + 6·53-s + 8·57-s + 12·59-s − 2·61-s + 2·63-s − 2·67-s − 12·69-s + 12·71-s + 2·73-s − 8·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.917·19-s − 0.872·21-s + 1.25·23-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.05·57-s + 1.56·59-s − 0.256·61-s + 0.251·63-s − 0.244·67-s − 1.44·69-s + 1.42·71-s + 0.234·73-s − 0.900·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28900\)    =    \(2^{2} \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{28900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 28900,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.225385985$
$L(\frac12)$  $\approx$  $1.225385985$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
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

−15.16087907793683, −14.70238880045124, −14.24657834649034, −13.53054146249441, −12.85051418263348, −12.54820926646322, −11.75987009959583, −11.53263167987284, −10.85080568098181, −10.63275038722811, −9.894986421571742, −9.211241408412817, −8.616399597807497, −8.058415744982228, −7.317945298954099, −6.827134258899572, −6.243141716851517, −5.395326750044494, −5.290555301495716, −4.456836757467861, −3.986484088738397, −2.879392482366463, −2.234973976227652, −1.284729040855391, −0.4951237445299724, 0.4951237445299724, 1.284729040855391, 2.234973976227652, 2.879392482366463, 3.986484088738397, 4.456836757467861, 5.290555301495716, 5.395326750044494, 6.243141716851517, 6.827134258899572, 7.317945298954099, 8.058415744982228, 8.616399597807497, 9.211241408412817, 9.894986421571742, 10.63275038722811, 10.85080568098181, 11.53263167987284, 11.75987009959583, 12.54820926646322, 12.85051418263348, 13.53054146249441, 14.24657834649034, 14.70238880045124, 15.16087907793683

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