Properties

Degree $2$
Conductor $227430$
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 − 5-s − 7-s − 8-s + 10-s + 5·11-s − 3·13-s + 14-s + 16-s + 5·17-s − 20-s − 5·22-s − 7·23-s + 25-s + 3·26-s − 28-s − 3·29-s − 6·31-s − 32-s − 5·34-s + 35-s + 2·37-s + 40-s + 41-s + 6·43-s + 5·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.50·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.223·20-s − 1.06·22-s − 1.45·23-s + 1/5·25-s + 0.588·26-s − 0.188·28-s − 0.557·29-s − 1.07·31-s − 0.176·32-s − 0.857·34-s + 0.169·35-s + 0.328·37-s + 0.158·40-s + 0.156·41-s + 0.914·43-s + 0.753·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(227430\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{227430} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 227430,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
19 \( 1 \)
good11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−12.97745316116617, −12.42146698799283, −12.28544973951840, −11.75245472462486, −11.34922950297279, −10.87928393739384, −10.20144887173544, −9.852543648242180, −9.447081064142923, −9.032638691856154, −8.570086504961764, −7.798098031397960, −7.606696914554328, −7.180502405076705, −6.525793206097843, −6.009843739465960, −5.743838403574130, −4.879982372650098, −4.318223205627133, −3.728613414515680, −3.365196555415279, −2.709739353490419, −1.889661263671838, −1.499738296615964, −0.6886786138371871, 0, 0.6886786138371871, 1.499738296615964, 1.889661263671838, 2.709739353490419, 3.365196555415279, 3.728613414515680, 4.318223205627133, 4.879982372650098, 5.743838403574130, 6.009843739465960, 6.525793206097843, 7.180502405076705, 7.606696914554328, 7.798098031397960, 8.570086504961764, 9.032638691856154, 9.447081064142923, 9.852543648242180, 10.20144887173544, 10.87928393739384, 11.34922950297279, 11.75245472462486, 12.28544973951840, 12.42146698799283, 12.97745316116617

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