Properties

Label 2-229320-1.1-c1-0-115
Degree $2$
Conductor $229320$
Sign $-1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
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  − 5-s + 5·11-s + 13-s − 3·17-s + 3·23-s + 25-s + 6·29-s + 2·31-s − 3·37-s − 41-s + 2·43-s − 6·47-s + 10·53-s − 5·55-s − 59-s + 2·61-s − 65-s + 13·67-s − 11·73-s − 5·79-s + 2·83-s + 3·85-s + 89-s − 5·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.50·11-s + 0.277·13-s − 0.727·17-s + 0.625·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s − 0.493·37-s − 0.156·41-s + 0.304·43-s − 0.875·47-s + 1.37·53-s − 0.674·55-s − 0.130·59-s + 0.256·61-s − 0.124·65-s + 1.58·67-s − 1.28·73-s − 0.562·79-s + 0.219·83-s + 0.325·85-s + 0.105·89-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.18282833463806, −12.64866894069602, −12.09323933301547, −11.85015404740657, −11.31354063484942, −10.97591124213490, −10.41148237997158, −9.851611232696755, −9.417195772357357, −8.813129986784562, −8.569416599233115, −8.124014900128465, −7.363058747103814, −6.891146605392180, −6.592398666130338, −6.129277691027596, −5.403078439559003, −4.893610262602373, −4.229649924819076, −4.003256124741693, −3.339539856631970, −2.783680921995459, −2.109296340989306, −1.322693555985191, −0.9248529314903778, 0, 0.9248529314903778, 1.322693555985191, 2.109296340989306, 2.783680921995459, 3.339539856631970, 4.003256124741693, 4.229649924819076, 4.893610262602373, 5.403078439559003, 6.129277691027596, 6.592398666130338, 6.891146605392180, 7.363058747103814, 8.124014900128465, 8.569416599233115, 8.813129986784562, 9.417195772357357, 9.851611232696755, 10.41148237997158, 10.97591124213490, 11.31354063484942, 11.85015404740657, 12.09323933301547, 12.64866894069602, 13.18282833463806

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