Properties

Label 2-229320-1.1-c1-0-111
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 13-s + 6·17-s − 4·19-s + 25-s + 2·29-s + 8·31-s + 2·37-s + 6·41-s − 4·43-s − 8·47-s + 10·53-s + 4·59-s − 14·61-s − 65-s + 4·67-s − 12·71-s + 10·73-s + 8·79-s + 12·83-s − 6·85-s + 14·89-s + 4·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.277·13-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1.37·53-s + 0.520·59-s − 1.79·61-s − 0.124·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s + 1.31·83-s − 0.650·85-s + 1.48·89-s + 0.410·95-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: \(1\)
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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.24665317140712, −12.50585607905678, −12.30509332074318, −11.76707524927366, −11.44732713606222, −10.73002777991655, −10.40634576534587, −10.03092850835901, −9.339836397759339, −9.031466631814791, −8.245591651593203, −8.037322682352058, −7.668436180273515, −6.935307287354950, −6.413127352383332, −6.139918067612036, −5.348617370389667, −5.002189037795509, −4.299917814499977, −3.918367710875544, −3.287276052367336, −2.784424767323495, −2.186972436223633, −1.313709980292491, −0.8771160115161669, 0, 0.8771160115161669, 1.313709980292491, 2.186972436223633, 2.784424767323495, 3.287276052367336, 3.918367710875544, 4.299917814499977, 5.002189037795509, 5.348617370389667, 6.139918067612036, 6.413127352383332, 6.935307287354950, 7.668436180273515, 8.037322682352058, 8.245591651593203, 9.031466631814791, 9.339836397759339, 10.03092850835901, 10.40634576534587, 10.73002777991655, 11.44732713606222, 11.76707524927366, 12.30509332074318, 12.50585607905678, 13.24665317140712

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