Properties

Label 2-4176-1.1-c1-0-47
Degree $2$
Conductor $4176$
Sign $-1$
Analytic cond. $33.3455$
Root an. cond. $5.77455$
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 − 2·7-s + 3·11-s − 13-s + 4·23-s − 4·25-s + 29-s − 3·31-s + 2·35-s − 8·37-s + 6·41-s + 5·43-s + 3·47-s − 3·49-s − 5·53-s − 3·55-s − 8·59-s + 65-s + 12·67-s + 6·71-s − 4·73-s − 6·77-s − 79-s − 12·83-s − 6·89-s + 2·91-s + 14·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 0.904·11-s − 0.277·13-s + 0.834·23-s − 4/5·25-s + 0.185·29-s − 0.538·31-s + 0.338·35-s − 1.31·37-s + 0.937·41-s + 0.762·43-s + 0.437·47-s − 3/7·49-s − 0.686·53-s − 0.404·55-s − 1.04·59-s + 0.124·65-s + 1.46·67-s + 0.712·71-s − 0.468·73-s − 0.683·77-s − 0.112·79-s − 1.31·83-s − 0.635·89-s + 0.209·91-s + 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4176\)    =    \(2^{4} \cdot 3^{2} \cdot 29\)
Sign: $-1$
Analytic conductor: \(33.3455\)
Root analytic conductor: \(5.77455\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4176,\ (\ :1/2),\ -1)\)

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 \( 1 \)
29 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.994610179369606995869884378763, −7.24961226348188201527304882529, −6.65320811852602296876789762362, −5.92169113770517864413828542879, −5.03803556130782721048021606314, −4.08187314982047767565464280256, −3.50289320210776225141646880508, −2.56353613863743857251285305435, −1.33307717435361558517907963551, 0, 1.33307717435361558517907963551, 2.56353613863743857251285305435, 3.50289320210776225141646880508, 4.08187314982047767565464280256, 5.03803556130782721048021606314, 5.92169113770517864413828542879, 6.65320811852602296876789762362, 7.24961226348188201527304882529, 7.994610179369606995869884378763

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