Properties

Label 2-1216-1.1-c3-0-26
Degree $2$
Conductor $1216$
Sign $1$
Analytic cond. $71.7463$
Root an. cond. $8.47032$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.67·3-s + 7.12·5-s + 0.511·7-s − 13.4·9-s + 17.5·11-s − 52.1·13-s − 26.1·15-s + 68.1·17-s − 19·19-s − 1.87·21-s − 48.8·23-s − 74.1·25-s + 148.·27-s + 87.1·29-s − 53.2·31-s − 64.5·33-s + 3.64·35-s + 199.·37-s + 191.·39-s − 372.·41-s + 416.·43-s − 96.2·45-s + 221.·47-s − 342.·49-s − 250.·51-s − 107.·53-s + 125.·55-s + ⋯
L(s)  = 1  − 0.707·3-s + 0.637·5-s + 0.0276·7-s − 0.499·9-s + 0.481·11-s − 1.11·13-s − 0.450·15-s + 0.972·17-s − 0.229·19-s − 0.0195·21-s − 0.442·23-s − 0.593·25-s + 1.06·27-s + 0.558·29-s − 0.308·31-s − 0.340·33-s + 0.0176·35-s + 0.887·37-s + 0.786·39-s − 1.41·41-s + 1.47·43-s − 0.318·45-s + 0.687·47-s − 0.999·49-s − 0.687·51-s − 0.279·53-s + 0.307·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $1$
Analytic conductor: \(71.7463\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.451176570\)
\(L(\frac12)\) \(\approx\) \(1.451176570\)
\(L(\frac{5}{2})\) 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 \( 1 + 19T \)
good3 \( 1 + 3.67T + 27T^{2} \)
5 \( 1 - 7.12T + 125T^{2} \)
7 \( 1 - 0.511T + 343T^{2} \)
11 \( 1 - 17.5T + 1.33e3T^{2} \)
13 \( 1 + 52.1T + 2.19e3T^{2} \)
17 \( 1 - 68.1T + 4.91e3T^{2} \)
23 \( 1 + 48.8T + 1.21e4T^{2} \)
29 \( 1 - 87.1T + 2.43e4T^{2} \)
31 \( 1 + 53.2T + 2.97e4T^{2} \)
37 \( 1 - 199.T + 5.06e4T^{2} \)
41 \( 1 + 372.T + 6.89e4T^{2} \)
43 \( 1 - 416.T + 7.95e4T^{2} \)
47 \( 1 - 221.T + 1.03e5T^{2} \)
53 \( 1 + 107.T + 1.48e5T^{2} \)
59 \( 1 - 155.T + 2.05e5T^{2} \)
61 \( 1 - 229.T + 2.26e5T^{2} \)
67 \( 1 - 580.T + 3.00e5T^{2} \)
71 \( 1 - 38.2T + 3.57e5T^{2} \)
73 \( 1 + 803.T + 3.89e5T^{2} \)
79 \( 1 - 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + 1.05e3T + 5.71e5T^{2} \)
89 \( 1 + 641.T + 7.04e5T^{2} \)
97 \( 1 - 19.9T + 9.12e5T^{2} \)
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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

−9.599796163184449611972879748781, −8.568061007236620338039829571204, −7.67199903585271551991318423402, −6.71137235735252631102967390518, −5.89364434631472611182568683835, −5.32311638799539424903578647471, −4.33129613830178354037903792424, −3.04099297433451327203649384537, −1.96198549648468818055924726568, −0.62872812969318314772714910639, 0.62872812969318314772714910639, 1.96198549648468818055924726568, 3.04099297433451327203649384537, 4.33129613830178354037903792424, 5.32311638799539424903578647471, 5.89364434631472611182568683835, 6.71137235735252631102967390518, 7.67199903585271551991318423402, 8.568061007236620338039829571204, 9.599796163184449611972879748781

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