Properties

Label 2-2736-1.1-c1-0-44
Degree $2$
Conductor $2736$
Sign $1$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 4·7-s − 4·11-s − 4·13-s − 6·17-s − 19-s − 6·23-s + 11·25-s − 2·29-s − 2·31-s + 16·35-s + 4·37-s + 6·41-s − 4·43-s − 2·47-s + 9·49-s + 6·53-s + 16·55-s − 4·59-s − 10·61-s + 16·65-s − 8·67-s − 2·73-s + 16·77-s − 14·79-s − 16·83-s + 24·85-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.51·7-s − 1.20·11-s − 1.10·13-s − 1.45·17-s − 0.229·19-s − 1.25·23-s + 11/5·25-s − 0.371·29-s − 0.359·31-s + 2.70·35-s + 0.657·37-s + 0.937·41-s − 0.609·43-s − 0.291·47-s + 9/7·49-s + 0.824·53-s + 2.15·55-s − 0.520·59-s − 1.28·61-s + 1.98·65-s − 0.977·67-s − 0.234·73-s + 1.82·77-s − 1.57·79-s − 1.75·83-s + 2.60·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 1)\)

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 \( 1 \)
19 \( 1 + T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−7.910424209436467226777304440060, −7.39841435492544052664442274969, −6.72950423472255228560372128637, −5.81636521987769670507563277506, −4.64105699328563059546993081215, −4.09576466163470275393774902456, −3.14093156035081707278369045534, −2.43899481601303924083267001618, 0, 0, 2.43899481601303924083267001618, 3.14093156035081707278369045534, 4.09576466163470275393774902456, 4.64105699328563059546993081215, 5.81636521987769670507563277506, 6.72950423472255228560372128637, 7.39841435492544052664442274969, 7.910424209436467226777304440060

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