Properties

Label 2-3192-1.1-c1-0-24
Degree $2$
Conductor $3192$
Sign $-1$
Analytic cond. $25.4882$
Root an. cond. $5.04858$
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  − 3-s − 4·5-s − 7-s + 9-s − 2·13-s + 4·15-s + 4·17-s + 19-s + 21-s + 4·23-s + 11·25-s − 27-s + 4·29-s + 4·35-s − 10·37-s + 2·39-s + 10·41-s + 4·43-s − 4·45-s − 6·47-s + 49-s − 4·51-s − 57-s + 4·59-s − 10·61-s − 63-s + 8·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.03·15-s + 0.970·17-s + 0.229·19-s + 0.218·21-s + 0.834·23-s + 11/5·25-s − 0.192·27-s + 0.742·29-s + 0.676·35-s − 1.64·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.596·45-s − 0.875·47-s + 1/7·49-s − 0.560·51-s − 0.132·57-s + 0.520·59-s − 1.28·61-s − 0.125·63-s + 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3192\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 19\)
Sign: $-1$
Analytic conductor: \(25.4882\)
Root analytic conductor: \(5.04858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3192,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
19 \( 1 - T \)
good5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.186808495375927710290517779319, −7.36380136998996332897085914758, −7.10245175705742542456471160146, −6.03924896184943598627006249770, −5.08084089218605196627138110393, −4.43021844461524779947464379669, −3.56418100801958708410840009819, −2.86583893479602789304544497029, −1.08895311619327121384054030083, 0, 1.08895311619327121384054030083, 2.86583893479602789304544497029, 3.56418100801958708410840009819, 4.43021844461524779947464379669, 5.08084089218605196627138110393, 6.03924896184943598627006249770, 7.10245175705742542456471160146, 7.36380136998996332897085914758, 8.186808495375927710290517779319

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