Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 \cdot 11 $
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-s − 4-s − 7-s + 3·8-s + 11-s − 6·13-s + 14-s − 16-s + 2·17-s + 4·19-s − 22-s + 6·26-s + 28-s + 2·29-s + 8·31-s − 5·32-s − 2·34-s − 6·37-s − 4·38-s − 10·41-s + 4·43-s − 44-s − 8·47-s + 49-s + 6·52-s + 6·53-s − 3·56-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 0.301·11-s − 1.66·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.213·22-s + 1.17·26-s + 0.188·28-s + 0.371·29-s + 1.43·31-s − 0.883·32-s − 0.342·34-s − 0.986·37-s − 0.648·38-s − 1.56·41-s + 0.609·43-s − 0.150·44-s − 1.16·47-s + 1/7·49-s + 0.832·52-s + 0.824·53-s − 0.400·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(17325\)    =    \(3^{2} \cdot 5^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{17325} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 17325,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−16.43209907042963, −15.54151531395562, −15.15136625440902, −14.36005344534890, −13.91626522343016, −13.54823554274355, −12.72402469450616, −12.13179247971653, −11.88381294794996, −10.93853799597437, −10.25427550073952, −9.771857152384081, −9.589842979748593, −8.787707132314724, −8.188740264542393, −7.648941855550916, −7.034542985566647, −6.499571372485118, −5.423960448748181, −5.001853494371032, −4.377039820590681, −3.483336365638079, −2.807154404558549, −1.823854392138734, −0.9152212210226658, 0, 0.9152212210226658, 1.823854392138734, 2.807154404558549, 3.483336365638079, 4.377039820590681, 5.001853494371032, 5.423960448748181, 6.499571372485118, 7.034542985566647, 7.648941855550916, 8.188740264542393, 8.787707132314724, 9.589842979748593, 9.771857152384081, 10.25427550073952, 10.93853799597437, 11.88381294794996, 12.13179247971653, 12.72402469450616, 13.54823554274355, 13.91626522343016, 14.36005344534890, 15.15136625440902, 15.54151531395562, 16.43209907042963

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