Properties

Label 2-1656-1.1-c1-0-21
Degree $2$
Conductor $1656$
Sign $-1$
Analytic cond. $13.2232$
Root an. cond. $3.63637$
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  − 2·7-s − 5·13-s + 6·17-s + 6·19-s − 23-s − 5·25-s − 9·29-s + 3·31-s − 8·37-s − 3·41-s − 8·43-s − 7·47-s − 3·49-s + 2·53-s − 4·59-s − 10·61-s + 8·67-s − 7·71-s + 9·73-s − 6·79-s + 14·83-s − 16·89-s + 10·91-s + 6·97-s − 6·101-s + 14·103-s − 14·107-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.38·13-s + 1.45·17-s + 1.37·19-s − 0.208·23-s − 25-s − 1.67·29-s + 0.538·31-s − 1.31·37-s − 0.468·41-s − 1.21·43-s − 1.02·47-s − 3/7·49-s + 0.274·53-s − 0.520·59-s − 1.28·61-s + 0.977·67-s − 0.830·71-s + 1.05·73-s − 0.675·79-s + 1.53·83-s − 1.69·89-s + 1.04·91-s + 0.609·97-s − 0.597·101-s + 1.37·103-s − 1.35·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1656\)    =    \(2^{3} \cdot 3^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(13.2232\)
Root analytic conductor: \(3.63637\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1656} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1656,\ (\ :1/2),\ -1)\)

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 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−9.220248871871571959783966327863, −7.955496038847795814418301587668, −7.48035014120651414389916926549, −6.62698149473987247879664007283, −5.57716866354923348134214900896, −5.03186685455612530303945885183, −3.66127566554473431534636647806, −3.03364632252564682099855997353, −1.69343828743407180715859546178, 0, 1.69343828743407180715859546178, 3.03364632252564682099855997353, 3.66127566554473431534636647806, 5.03186685455612530303945885183, 5.57716866354923348134214900896, 6.62698149473987247879664007283, 7.48035014120651414389916926549, 7.955496038847795814418301587668, 9.220248871871571959783966327863

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