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Properties

Label 2-76e2-1.1-c1-0-125
Degree $2$
Conductor $5776$
Sign $-1$
Analytic cond. $46.1215$
Root an. cond. $6.79128$
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 + 3·5-s − 2·9-s + 4·11-s − 5·13-s − 3·15-s − 5·17-s + 23-s + 4·25-s + 5·27-s + 3·29-s − 4·31-s − 4·33-s + 2·37-s + 5·39-s − 5·41-s + 11·43-s − 6·45-s + 5·47-s − 7·49-s + 5·51-s − 9·53-s + 12·55-s − 13·59-s − 61-s − 15·65-s + 5·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 2/3·9-s + 1.20·11-s − 1.38·13-s − 0.774·15-s − 1.21·17-s + 0.208·23-s + 4/5·25-s + 0.962·27-s + 0.557·29-s − 0.718·31-s − 0.696·33-s + 0.328·37-s + 0.800·39-s − 0.780·41-s + 1.67·43-s − 0.894·45-s + 0.729·47-s − 49-s + 0.700·51-s − 1.23·53-s + 1.61·55-s − 1.69·59-s − 0.128·61-s − 1.86·65-s + 0.610·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5776\)    =    \(2^{4} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(46.1215\)
Root analytic conductor: \(6.79128\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5776,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−7.59872473365232218068791948934, −6.83075044054572343025756669507, −6.23902962294964096714433558555, −5.75931851189139969458779396269, −4.91215892937544006745154073668, −4.34718693991791536259345999078, −3.02054133401327408268063812218, −2.28887187412560605052544112344, −1.40646902110043167226315148861, 0, 1.40646902110043167226315148861, 2.28887187412560605052544112344, 3.02054133401327408268063812218, 4.34718693991791536259345999078, 4.91215892937544006745154073668, 5.75931851189139969458779396269, 6.23902962294964096714433558555, 6.83075044054572343025756669507, 7.59872473365232218068791948934

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