Properties

Degree $2$
Conductor $42350$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 3·9-s − 6·13-s − 14-s + 16-s + 2·17-s − 3·18-s − 6·26-s − 28-s − 6·29-s + 8·31-s + 32-s + 2·34-s − 3·36-s + 10·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 6·52-s + 2·53-s − 56-s − 6·58-s − 8·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 1.17·26-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.832·52-s + 0.274·53-s − 0.133·56-s − 0.787·58-s − 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42350\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{42350} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 42350,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.82662078106542, −14.57257680348763, −13.98358591022275, −13.42310685691287, −12.92610959201452, −12.38592369923969, −11.90335956429882, −11.50641970733895, −10.96321971070093, −10.23399617411250, −9.722612589423903, −9.363351214934021, −8.539734659638513, −7.870634226968134, −7.548900111588559, −6.783225050277882, −6.252398580895870, −5.698956929482763, −5.089633438032180, −4.655159146843520, −3.850656645448166, −3.193624504926450, −2.575357287058195, −2.174647496205383, −0.9494134305738243, 0, 0.9494134305738243, 2.174647496205383, 2.575357287058195, 3.193624504926450, 3.850656645448166, 4.655159146843520, 5.089633438032180, 5.698956929482763, 6.252398580895870, 6.783225050277882, 7.548900111588559, 7.870634226968134, 8.539734659638513, 9.363351214934021, 9.722612589423903, 10.23399617411250, 10.96321971070093, 11.50641970733895, 11.90335956429882, 12.38592369923969, 12.92610959201452, 13.42310685691287, 13.98358591022275, 14.57257680348763, 14.82662078106542

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