Properties

Degree $2$
Conductor $105966$
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 − 7-s + 8-s + 2·10-s − 4·11-s + 6·13-s − 14-s + 16-s + 2·17-s + 4·19-s + 2·20-s − 4·22-s − 8·23-s − 25-s + 6·26-s − 28-s + 32-s + 2·34-s − 2·35-s + 10·37-s + 4·38-s + 2·40-s − 6·41-s + 4·43-s − 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s − 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 1.66·23-s − 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.176·32-s + 0.342·34-s − 0.338·35-s + 1.64·37-s + 0.648·38-s + 0.316·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105966\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 29^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{105966} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 105966,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
29 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.85660574310879, −13.36486917900637, −13.20874606673305, −12.65666221661443, −11.99117662730688, −11.62791543018648, −10.97857124481096, −10.52868905901519, −10.05833465541661, −9.667611048960616, −9.087191667682297, −8.390620844279905, −7.772857249694640, −7.624251666701913, −6.651973552612577, −6.084924110458726, −5.906713655173372, −5.443826945078380, −4.772614363865631, −4.080502087004270, −3.546037221344114, −2.951410618830526, −2.424667105371893, −1.671911632745683, −1.124952159352948, 0, 1.124952159352948, 1.671911632745683, 2.424667105371893, 2.951410618830526, 3.546037221344114, 4.080502087004270, 4.772614363865631, 5.443826945078380, 5.906713655173372, 6.084924110458726, 6.651973552612577, 7.624251666701913, 7.772857249694640, 8.390620844279905, 9.087191667682297, 9.667611048960616, 10.05833465541661, 10.52868905901519, 10.97857124481096, 11.62791543018648, 11.99117662730688, 12.65666221661443, 13.20874606673305, 13.36486917900637, 13.85660574310879

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