Properties

Label 2-31680-1.1-c1-0-106
Degree $2$
Conductor $31680$
Sign $-1$
Analytic cond. $252.966$
Root an. cond. $15.9049$
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 − 7-s + 11-s + 2·13-s + 5·17-s + 7·19-s + 6·23-s + 25-s − 29-s − 5·31-s − 35-s − 11·37-s − 2·41-s − 4·43-s − 6·47-s − 6·49-s − 53-s + 55-s − 10·59-s − 5·61-s + 2·65-s + 8·67-s − 9·71-s − 6·73-s − 77-s − 10·79-s − 6·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.301·11-s + 0.554·13-s + 1.21·17-s + 1.60·19-s + 1.25·23-s + 1/5·25-s − 0.185·29-s − 0.898·31-s − 0.169·35-s − 1.80·37-s − 0.312·41-s − 0.609·43-s − 0.875·47-s − 6/7·49-s − 0.137·53-s + 0.134·55-s − 1.30·59-s − 0.640·61-s + 0.248·65-s + 0.977·67-s − 1.06·71-s − 0.702·73-s − 0.113·77-s − 1.12·79-s − 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31680\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(252.966\)
Root analytic conductor: \(15.9049\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31680,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 - T \)
good7 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 17 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−15.48436439909503, −14.61369926368823, −14.31183815424475, −13.78431721665155, −13.21535870150880, −12.77039271031397, −12.16002375114736, −11.62653253903740, −11.11372474794752, −10.46193615371278, −9.869223192958404, −9.526011692665329, −8.862421109181866, −8.424769140117076, −7.449705447246313, −7.264005171118320, −6.460517619202224, −5.904605513738320, −5.214452899713103, −4.937516286737482, −3.756821631244974, −3.296072641046842, −2.853322164727887, −1.503504096663033, −1.332398872261024, 0, 1.332398872261024, 1.503504096663033, 2.853322164727887, 3.296072641046842, 3.756821631244974, 4.937516286737482, 5.214452899713103, 5.904605513738320, 6.460517619202224, 7.264005171118320, 7.449705447246313, 8.424769140117076, 8.862421109181866, 9.526011692665329, 9.869223192958404, 10.46193615371278, 11.11372474794752, 11.62653253903740, 12.16002375114736, 12.77039271031397, 13.21535870150880, 13.78431721665155, 14.31183815424475, 14.61369926368823, 15.48436439909503

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