Properties

Label 2-229320-1.1-c1-0-60
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 + 17-s − 6·19-s − 3·23-s + 25-s + 6·29-s − 4·31-s − 11·37-s + 9·41-s + 6·43-s − 4·47-s + 6·53-s + 5·55-s − 9·59-s − 10·61-s − 65-s − 3·67-s − 8·71-s + 73-s + 15·79-s − 8·83-s − 85-s − 89-s + 6·95-s − 5·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.50·11-s + 0.277·13-s + 0.242·17-s − 1.37·19-s − 0.625·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 1.80·37-s + 1.40·41-s + 0.914·43-s − 0.583·47-s + 0.824·53-s + 0.674·55-s − 1.17·59-s − 1.28·61-s − 0.124·65-s − 0.366·67-s − 0.949·71-s + 0.117·73-s + 1.68·79-s − 0.878·83-s − 0.108·85-s − 0.105·89-s + 0.615·95-s − 0.507·97-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 \( 1 - T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
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

−13.06332671700308, −12.67054053486202, −12.24789424664369, −11.96575283712541, −11.11951584432555, −10.79135033559115, −10.43692722771860, −10.16652123047661, −9.196123772212038, −9.078581479522984, −8.232533135193680, −8.070775178426815, −7.633280207012088, −6.946460665494430, −6.591724838558293, −5.807427075454585, −5.582521850377178, −4.897969917092591, −4.316405648233406, −4.026101135195953, −3.127201824904379, −2.848331866985335, −2.110380440520705, −1.588260918321288, −0.5943222775978844, 0, 0.5943222775978844, 1.588260918321288, 2.110380440520705, 2.848331866985335, 3.127201824904379, 4.026101135195953, 4.316405648233406, 4.897969917092591, 5.582521850377178, 5.807427075454585, 6.591724838558293, 6.946460665494430, 7.633280207012088, 8.070775178426815, 8.232533135193680, 9.078581479522984, 9.196123772212038, 10.16652123047661, 10.43692722771860, 10.79135033559115, 11.11951584432555, 11.96575283712541, 12.24789424664369, 12.67054053486202, 13.06332671700308

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