Properties

Label 2-229320-1.1-c1-0-43
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 − 4·11-s − 13-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 6·29-s + 4·31-s + 6·37-s − 2·41-s + 12·43-s + 8·47-s − 14·53-s + 4·55-s + 12·59-s + 2·61-s + 65-s − 8·71-s + 2·73-s + 8·79-s + 12·83-s + 2·85-s + 6·89-s − 4·95-s + 10·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s − 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.312·41-s + 1.82·43-s + 1.16·47-s − 1.92·53-s + 0.539·55-s + 1.56·59-s + 0.256·61-s + 0.124·65-s − 0.949·71-s + 0.234·73-s + 0.900·79-s + 1.31·83-s + 0.216·85-s + 0.635·89-s − 0.410·95-s + 1.01·97-s + 0.0995·101-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

\(L(1)\) \(\approx\) \(2.252457572\)
\(L(\frac12)\) \(\approx\) \(2.252457572\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.02712629295488, −12.34159115543904, −12.06285962513603, −11.62901372724876, −10.92397925852459, −10.76637817232819, −10.04573349733879, −9.829987658348902, −9.151804791620102, −8.698758367264711, −8.066835068826854, −7.735150928710629, −7.451354222471405, −6.718131428653023, −6.239468449546756, −5.702749573257650, −5.158431026814609, −4.623606339657992, −4.259907553543816, −3.543032981210680, −2.927463865655272, −2.497522722448770, −1.955961847422673, −0.8999200611896808, −0.5174869174387965, 0.5174869174387965, 0.8999200611896808, 1.955961847422673, 2.497522722448770, 2.927463865655272, 3.543032981210680, 4.259907553543816, 4.623606339657992, 5.158431026814609, 5.702749573257650, 6.239468449546756, 6.718131428653023, 7.451354222471405, 7.735150928710629, 8.066835068826854, 8.698758367264711, 9.151804791620102, 9.829987658348902, 10.04573349733879, 10.76637817232819, 10.92397925852459, 11.62901372724876, 12.06285962513603, 12.34159115543904, 13.02712629295488

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