Properties

Label 2-8550-1.1-c1-0-120
Degree $2$
Conductor $8550$
Sign $-1$
Analytic cond. $68.2720$
Root an. cond. $8.26269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4·7-s + 8-s + 4·11-s + 2·13-s − 4·14-s + 16-s − 2·17-s − 19-s + 4·22-s − 8·23-s + 2·26-s − 4·28-s − 6·29-s + 4·31-s + 32-s − 2·34-s + 10·37-s − 38-s + 2·41-s − 12·43-s + 4·44-s − 8·46-s + 9·49-s + 2·52-s + 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 1.20·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.229·19-s + 0.852·22-s − 1.66·23-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 1.64·37-s − 0.162·38-s + 0.312·41-s − 1.82·43-s + 0.603·44-s − 1.17·46-s + 9/7·49-s + 0.277·52-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(68.2720\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−7.15770912822291761947965035369, −6.48801873778727791621256291858, −6.19563343316073825191709338723, −5.55305678482459290140237738191, −4.29915055845805882478815828671, −3.96564951081572590139725777543, −3.24728229749007396311542791168, −2.40176698563740779955778170375, −1.38369596314356637117144031912, 0, 1.38369596314356637117144031912, 2.40176698563740779955778170375, 3.24728229749007396311542791168, 3.96564951081572590139725777543, 4.29915055845805882478815828671, 5.55305678482459290140237738191, 6.19563343316073825191709338723, 6.48801873778727791621256291858, 7.15770912822291761947965035369

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