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 + 2·13-s + 14-s + 16-s − 2·17-s − 20-s + 4·23-s + 25-s − 2·26-s − 28-s − 6·29-s + 8·31-s − 32-s + 2·34-s + 35-s + 2·37-s + 40-s − 2·41-s + 4·43-s − 4·46-s + 4·47-s + 49-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 + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.328·37-s + 0.158·40-s − 0.312·41-s + 0.609·43-s − 0.589·46-s + 0.583·47-s + 1/7·49-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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.13170520981213, −12.66051408899272, −12.21946781659044, −11.55085452410858, −11.34830284135962, −10.87938579416999, −10.26353817914121, −10.02450082711602, −9.306278799809454, −8.811219878226818, −8.745934613571542, −7.891540473179405, −7.606567234343673, −7.063886375516832, −6.592138217759356, −6.022922396604931, −5.711184299815840, −4.792286317268179, −4.472230024561113, −3.734033073464149, −3.236226113637453, −2.715822348200201, −2.070017927396039, −1.326524295141274, −0.7400825798078273, 0, 0.7400825798078273, 1.326524295141274, 2.070017927396039, 2.715822348200201, 3.236226113637453, 3.734033073464149, 4.472230024561113, 4.792286317268179, 5.711184299815840, 6.022922396604931, 6.592138217759356, 7.063886375516832, 7.606567234343673, 7.891540473179405, 8.745934613571542, 8.811219878226818, 9.306278799809454, 10.02450082711602, 10.26353817914121, 10.87938579416999, 11.34830284135962, 11.55085452410858, 12.21946781659044, 12.66051408899272, 13.13170520981213

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