Properties

Label 2-229320-1.1-c1-0-79
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 − 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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
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

−13.17848706715213, −12.78579343774663, −12.24006066152160, −11.91481477283589, −11.16470956515041, −10.78367797885527, −10.34305170012423, −10.00173697120223, −9.560502241809817, −8.841323418871745, −8.406337726728855, −8.095238009731977, −7.548644413041202, −6.829412838404981, −6.571376608921510, −6.011192115612440, −5.309903533182563, −5.011418883379838, −4.552882635562069, −3.898829312942607, −3.040152527336644, −2.801363930044389, −2.124896420310321, −1.626243837748620, −0.6918348499557924, 0, 0.6918348499557924, 1.626243837748620, 2.124896420310321, 2.801363930044389, 3.040152527336644, 3.898829312942607, 4.552882635562069, 5.011418883379838, 5.309903533182563, 6.011192115612440, 6.571376608921510, 6.829412838404981, 7.548644413041202, 8.095238009731977, 8.406337726728855, 8.841323418871745, 9.560502241809817, 10.00173697120223, 10.34305170012423, 10.78367797885527, 11.16470956515041, 11.91481477283589, 12.24006066152160, 12.78579343774663, 13.17848706715213

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