Properties

Degree $2$
Conductor $1989$
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 + 2·5-s − 3·8-s + 2·10-s − 4·11-s + 13-s − 16-s − 17-s − 4·19-s − 2·20-s − 4·22-s − 25-s + 26-s + 2·29-s − 8·31-s + 5·32-s − 34-s − 2·37-s − 4·38-s − 6·40-s − 2·41-s − 4·43-s + 4·44-s − 8·47-s − 7·49-s − 50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 0.632·10-s − 1.20·11-s + 0.277·13-s − 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.196·26-s + 0.371·29-s − 1.43·31-s + 0.883·32-s − 0.171·34-s − 0.328·37-s − 0.648·38-s − 0.948·40-s − 0.312·41-s − 0.609·43-s + 0.603·44-s − 1.16·47-s − 49-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1989 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1989 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1989\)    =    \(3^{2} \cdot 13 \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1989} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1989,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 - T \)
17 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.79502710826905, −18.92789742152318, −18.33762532714445, −17.81313717399689, −17.34447332361365, −16.39163590891467, −15.75404806770464, −14.84810549749875, −14.53076929065959, −13.49985021297613, −13.27985807853093, −12.75940789043372, −11.88330814709071, −10.95892336862070, −10.20851609531753, −9.620346754474105, −8.729727343516887, −8.185596227926961, −7.016076746686810, −6.108712628831280, −5.480922382799510, −4.839287632717801, −3.864225280932047, −2.874210053977077, −1.875569136657664, 0, 1.875569136657664, 2.874210053977077, 3.864225280932047, 4.839287632717801, 5.480922382799510, 6.108712628831280, 7.016076746686810, 8.185596227926961, 8.729727343516887, 9.620346754474105, 10.20851609531753, 10.95892336862070, 11.88330814709071, 12.75940789043372, 13.27985807853093, 13.49985021297613, 14.53076929065959, 14.84810549749875, 15.75404806770464, 16.39163590891467, 17.34447332361365, 17.81313717399689, 18.33762532714445, 18.92789742152318, 19.79502710826905

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