Properties

Label 2-8664-1.1-c1-0-166
Degree $2$
Conductor $8664$
Sign $-1$
Analytic cond. $69.1823$
Root an. cond. $8.31759$
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-s + 0.694·5-s + 0.305·7-s + 9-s + 4.82·11-s − 13-s + 0.694·15-s − 7.51·17-s + 0.305·21-s + 0.694·23-s − 4.51·25-s + 27-s − 10.1·29-s + 1.82·31-s + 4.82·33-s + 0.212·35-s − 6.51·37-s − 39-s − 5.38·41-s + 3.69·43-s + 0.694·45-s − 6·47-s − 6.90·49-s − 7.51·51-s − 5.43·53-s + 3.34·55-s + 4.08·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.310·5-s + 0.115·7-s + 0.333·9-s + 1.45·11-s − 0.277·13-s + 0.179·15-s − 1.82·17-s + 0.0666·21-s + 0.144·23-s − 0.903·25-s + 0.192·27-s − 1.88·29-s + 0.327·31-s + 0.839·33-s + 0.0358·35-s − 1.07·37-s − 0.160·39-s − 0.841·41-s + 0.563·43-s + 0.103·45-s − 0.875·47-s − 0.986·49-s − 1.05·51-s − 0.746·53-s + 0.451·55-s + 0.531·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8664\)    =    \(2^{3} \cdot 3 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(69.1823\)
Root analytic conductor: \(8.31759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8664,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
19 \( 1 \)
good5 \( 1 - 0.694T + 5T^{2} \)
7 \( 1 - 0.305T + 7T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + 7.51T + 17T^{2} \)
23 \( 1 - 0.694T + 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 - 1.82T + 31T^{2} \)
37 \( 1 + 6.51T + 37T^{2} \)
41 \( 1 + 5.38T + 41T^{2} \)
43 \( 1 - 3.69T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 5.43T + 53T^{2} \)
59 \( 1 - 4.08T + 59T^{2} \)
61 \( 1 - 0.389T + 61T^{2} \)
67 \( 1 - 7.82T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + 4.38T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 0.739T + 83T^{2} \)
89 \( 1 - 0.822T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−7.32111366365350074581422153174, −6.79078545380025192134497466910, −6.20847736599672451544253035399, −5.33398173846403903220510429581, −4.42712243405257729728345888730, −3.92397261493310377912322884568, −3.10175522928482241903127756479, −2.00201159338040773759171775047, −1.60909202142343944445916908517, 0, 1.60909202142343944445916908517, 2.00201159338040773759171775047, 3.10175522928482241903127756479, 3.92397261493310377912322884568, 4.42712243405257729728345888730, 5.33398173846403903220510429581, 6.20847736599672451544253035399, 6.79078545380025192134497466910, 7.32111366365350074581422153174

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