Properties

Label 2-229320-1.1-c1-0-25
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 5·11-s + 13-s + 17-s + 4·19-s + 23-s + 25-s + 6·29-s − 6·31-s − 7·37-s + 7·41-s + 6·43-s + 10·47-s − 6·53-s + 5·55-s − 7·59-s + 2·61-s − 65-s − 67-s + 8·71-s + 73-s − 11·79-s − 6·83-s − 85-s − 15·89-s − 4·95-s + 7·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.50·11-s + 0.277·13-s + 0.242·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s − 1.07·31-s − 1.15·37-s + 1.09·41-s + 0.914·43-s + 1.45·47-s − 0.824·53-s + 0.674·55-s − 0.911·59-s + 0.256·61-s − 0.124·65-s − 0.122·67-s + 0.949·71-s + 0.117·73-s − 1.23·79-s − 0.658·83-s − 0.108·85-s − 1.58·89-s − 0.410·95-s + 0.710·97-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: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.846471413\)
\(L(\frac12)\) \(\approx\) \(1.846471413\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−12.68726038632212, −12.58417045172973, −12.14624889009574, −11.43273377951532, −11.04278142483489, −10.67908554238174, −10.19539368022071, −9.740534694716799, −9.105156715512003, −8.721680406140140, −8.121821290112863, −7.760690303667628, −7.197959395870100, −7.032134514658131, −6.027131880149181, −5.736471563831798, −5.220177184959307, −4.679024109968485, −4.201285844919448, −3.441072021076529, −3.058061095141768, −2.521904773708592, −1.845469063186185, −1.016446995246845, −0.4287932100707081, 0.4287932100707081, 1.016446995246845, 1.845469063186185, 2.521904773708592, 3.058061095141768, 3.441072021076529, 4.201285844919448, 4.679024109968485, 5.220177184959307, 5.736471563831798, 6.027131880149181, 7.032134514658131, 7.197959395870100, 7.760690303667628, 8.121821290112863, 8.721680406140140, 9.105156715512003, 9.740534694716799, 10.19539368022071, 10.67908554238174, 11.04278142483489, 11.43273377951532, 12.14624889009574, 12.58417045172973, 12.68726038632212

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