Properties

Label 2-229320-1.1-c1-0-64
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 + 6·17-s + 4·23-s + 25-s + 6·29-s + 8·31-s − 2·37-s + 10·41-s − 4·43-s + 8·47-s + 2·53-s − 4·55-s + 4·59-s − 14·61-s − 65-s − 12·67-s + 8·71-s + 10·73-s − 4·83-s + 6·85-s + 10·89-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s − 0.277·13-s + 1.45·17-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 0.274·53-s − 0.539·55-s + 0.520·59-s − 1.79·61-s − 0.124·65-s − 1.46·67-s + 0.949·71-s + 1.17·73-s − 0.439·83-s + 0.650·85-s + 1.05·89-s + 0.203·97-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: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.582666576\)
\(L(\frac12)\) \(\approx\) \(3.582666576\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.90483540044156, −12.44339128456093, −12.10145970424646, −11.65369866694682, −10.88963114151221, −10.55212974276742, −10.18548259691748, −9.768143064972959, −9.232468794550690, −8.690901841036254, −8.200053725261332, −7.605792246930202, −7.466903806064692, −6.684156831854949, −6.163498257521016, −5.720601427789901, −5.163401672205863, −4.796188043000925, −4.258048498293152, −3.393638286878159, −2.913986457084212, −2.570969280468957, −1.850709394423424, −0.9953854968025348, −0.6194880419421708, 0.6194880419421708, 0.9953854968025348, 1.850709394423424, 2.570969280468957, 2.913986457084212, 3.393638286878159, 4.258048498293152, 4.796188043000925, 5.163401672205863, 5.720601427789901, 6.163498257521016, 6.684156831854949, 7.466903806064692, 7.605792246930202, 8.200053725261332, 8.690901841036254, 9.232468794550690, 9.768143064972959, 10.18548259691748, 10.55212974276742, 10.88963114151221, 11.65369866694682, 12.10145970424646, 12.44339128456093, 12.90483540044156

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