Properties

Label 2-1638-1.1-c1-0-23
Degree $2$
Conductor $1638$
Sign $-1$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  − 2-s + 4-s + 7-s − 8-s + 13-s − 14-s + 16-s − 6·17-s − 4·19-s − 6·23-s − 5·25-s − 26-s + 28-s + 8·31-s − 32-s + 6·34-s + 2·37-s + 4·38-s − 4·43-s + 6·46-s − 6·47-s + 49-s + 5·50-s + 52-s − 56-s − 6·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 1.25·23-s − 25-s − 0.196·26-s + 0.188·28-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.648·38-s − 0.609·43-s + 0.884·46-s − 0.875·47-s + 1/7·49-s + 0.707·50-s + 0.138·52-s − 0.133·56-s − 0.781·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
7 \( 1 - T \)
13 \( 1 - T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−8.833123066969410272291086657384, −8.315380262939132711743531410580, −7.59113649640036618050579447442, −6.52107659919036530841891307507, −6.05514828155563894984871537986, −4.73802182050113221543979801436, −3.94161510269754508782544806387, −2.55291695292451487866541530230, −1.66895190682184687500782568597, 0, 1.66895190682184687500782568597, 2.55291695292451487866541530230, 3.94161510269754508782544806387, 4.73802182050113221543979801436, 6.05514828155563894984871537986, 6.52107659919036530841891307507, 7.59113649640036618050579447442, 8.315380262939132711743531410580, 8.833123066969410272291086657384

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