Properties

Label 2-96e2-1.1-c1-0-64
Degree $2$
Conductor $9216$
Sign $-1$
Analytic cond. $73.5901$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46·5-s − 1.41·7-s − 4.89·11-s − 1.41·13-s + 4.89·17-s − 6·19-s + 6.92·23-s + 6.99·25-s + 3.46·29-s + 1.41·31-s + 4.89·35-s + 9.89·37-s + 4.89·41-s − 6·43-s + 6.92·47-s − 5·49-s − 3.46·53-s + 16.9·55-s − 9.79·59-s + 7.07·61-s + 4.89·65-s + 8·67-s + 13.8·71-s − 12·73-s + 6.92·77-s − 15.5·79-s + 14.6·83-s + ⋯
L(s)  = 1  − 1.54·5-s − 0.534·7-s − 1.47·11-s − 0.392·13-s + 1.18·17-s − 1.37·19-s + 1.44·23-s + 1.39·25-s + 0.643·29-s + 0.254·31-s + 0.828·35-s + 1.62·37-s + 0.765·41-s − 0.914·43-s + 1.01·47-s − 0.714·49-s − 0.475·53-s + 2.28·55-s − 1.27·59-s + 0.905·61-s + 0.607·65-s + 0.977·67-s + 1.64·71-s − 1.40·73-s + 0.789·77-s − 1.75·79-s + 1.61·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9216\)    =    \(2^{10} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(73.5901\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9216,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good5 \( 1 + 3.46T + 5T^{2} \)
7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 - 9.89T + 37T^{2} \)
41 \( 1 - 4.89T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 3.46T + 53T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 - 7.07T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 - 9.79T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−7.57021422842504162177172984828, −6.82324756184680800753763020481, −6.07936766310609146070048378405, −5.11233659923648313312959291066, −4.61979646511995695550295598476, −3.80165549710596219619860832081, −3.03333064259171931554732589402, −2.51279105978183932680662705842, −0.900013525849858742360923892661, 0, 0.900013525849858742360923892661, 2.51279105978183932680662705842, 3.03333064259171931554732589402, 3.80165549710596219619860832081, 4.61979646511995695550295598476, 5.11233659923648313312959291066, 6.07936766310609146070048378405, 6.82324756184680800753763020481, 7.57021422842504162177172984828

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