Properties

Label 2-3192-1.1-c1-0-41
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 + 7-s + 9-s + 2·11-s − 4·13-s + 19-s − 21-s − 6·23-s − 5·25-s − 27-s + 2·29-s + 4·31-s − 2·33-s + 2·37-s + 4·39-s + 2·41-s − 8·43-s − 8·47-s + 49-s − 6·53-s − 57-s − 4·59-s + 14·61-s + 63-s − 2·67-s + 6·69-s − 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.229·19-s − 0.218·21-s − 1.25·23-s − 25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.348·33-s + 0.328·37-s + 0.640·39-s + 0.312·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.132·57-s − 0.520·59-s + 1.79·61-s + 0.125·63-s − 0.244·67-s + 0.722·69-s − 0.702·73-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 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−8.126728822965856872245344268629, −7.61449736836411210829257604926, −6.68471916605767197977812703357, −6.08600051705823631580745227099, −5.17388322184790843889986275291, −4.52107706135484841599095676545, −3.67269445153567228289988183780, −2.43859040498123804703949075559, −1.44599544060078995492712723228, 0, 1.44599544060078995492712723228, 2.43859040498123804703949075559, 3.67269445153567228289988183780, 4.52107706135484841599095676545, 5.17388322184790843889986275291, 6.08600051705823631580745227099, 6.68471916605767197977812703357, 7.61449736836411210829257604926, 8.126728822965856872245344268629

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