Properties

Label 2-229320-1.1-c1-0-108
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 + 2·11-s − 13-s − 17-s + 3·19-s − 4·23-s + 25-s − 2·29-s − 4·31-s − 41-s + 5·43-s + 6·47-s + 14·53-s + 2·55-s − 10·59-s − 4·61-s − 65-s + 2·67-s − 9·71-s + 7·73-s − 15·79-s + 9·83-s − 85-s + 11·89-s + 3·95-s − 5·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.603·11-s − 0.277·13-s − 0.242·17-s + 0.688·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.156·41-s + 0.762·43-s + 0.875·47-s + 1.92·53-s + 0.269·55-s − 1.30·59-s − 0.512·61-s − 0.124·65-s + 0.244·67-s − 1.06·71-s + 0.819·73-s − 1.68·79-s + 0.987·83-s − 0.108·85-s + 1.16·89-s + 0.307·95-s − 0.507·97-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.28384858172323, −12.53434655602807, −12.32557190325175, −11.80026250563191, −11.33607281044885, −10.83128441209718, −10.34257472300604, −9.877246051381384, −9.447681523244501, −8.888786261759943, −8.718152694951937, −7.828136321144800, −7.501591512228158, −7.022822433216539, −6.470148611874367, −5.833994149604428, −5.650320672730039, −4.942905075958892, −4.375468128845549, −3.866702737650653, −3.349787060563413, −2.621455123310886, −2.128066294515858, −1.507487556544676, −0.8606413449184059, 0, 0.8606413449184059, 1.507487556544676, 2.128066294515858, 2.621455123310886, 3.349787060563413, 3.866702737650653, 4.375468128845549, 4.942905075958892, 5.650320672730039, 5.833994149604428, 6.470148611874367, 7.022822433216539, 7.501591512228158, 7.828136321144800, 8.718152694951937, 8.888786261759943, 9.447681523244501, 9.877246051381384, 10.34257472300604, 10.83128441209718, 11.33607281044885, 11.80026250563191, 12.32557190325175, 12.53434655602807, 13.28384858172323

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