Properties

Label 2-2e3-1.1-c9-0-1
Degree $2$
Conductor $8$
Sign $-1$
Analytic cond. $4.12028$
Root an. cond. $2.02984$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 60·3-s − 2.07e3·5-s − 4.34e3·7-s − 1.60e4·9-s + 9.36e4·11-s − 1.22e4·13-s + 1.24e5·15-s − 3.19e5·17-s − 5.53e5·19-s + 2.60e5·21-s − 7.12e5·23-s + 2.34e6·25-s + 2.14e6·27-s + 2.07e6·29-s − 6.42e6·31-s − 5.61e6·33-s + 9.00e6·35-s − 1.81e7·37-s + 7.34e5·39-s + 9.03e6·41-s + 1.95e7·43-s + 3.33e7·45-s − 1.84e7·47-s − 2.14e7·49-s + 1.91e7·51-s + 1.02e7·53-s − 1.94e8·55-s + ⋯
L(s)  = 1  − 0.427·3-s − 1.48·5-s − 0.683·7-s − 0.817·9-s + 1.92·11-s − 0.118·13-s + 0.634·15-s − 0.928·17-s − 0.974·19-s + 0.292·21-s − 0.531·23-s + 1.20·25-s + 0.777·27-s + 0.545·29-s − 1.24·31-s − 0.824·33-s + 1.01·35-s − 1.59·37-s + 0.0508·39-s + 0.499·41-s + 0.874·43-s + 1.21·45-s − 0.552·47-s − 0.532·49-s + 0.396·51-s + 0.178·53-s − 2.86·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-1$
Analytic conductor: \(4.12028\)
Root analytic conductor: \(2.02984\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 20 p T + p^{9} T^{2} \)
5 \( 1 + 2074 T + p^{9} T^{2} \)
7 \( 1 + 4344 T + p^{9} T^{2} \)
11 \( 1 - 93644 T + p^{9} T^{2} \)
13 \( 1 + 12242 T + p^{9} T^{2} \)
17 \( 1 + 319598 T + p^{9} T^{2} \)
19 \( 1 + 553516 T + p^{9} T^{2} \)
23 \( 1 + 712936 T + p^{9} T^{2} \)
29 \( 1 - 2075838 T + p^{9} T^{2} \)
31 \( 1 + 6420448 T + p^{9} T^{2} \)
37 \( 1 + 18197754 T + p^{9} T^{2} \)
41 \( 1 - 9033834 T + p^{9} T^{2} \)
43 \( 1 - 19594732 T + p^{9} T^{2} \)
47 \( 1 + 18484176 T + p^{9} T^{2} \)
53 \( 1 - 10255766 T + p^{9} T^{2} \)
59 \( 1 - 121666556 T + p^{9} T^{2} \)
61 \( 1 + 45948962 T + p^{9} T^{2} \)
67 \( 1 - 50535428 T + p^{9} T^{2} \)
71 \( 1 - 267044680 T + p^{9} T^{2} \)
73 \( 1 + 176213366 T + p^{9} T^{2} \)
79 \( 1 + 269685680 T + p^{9} T^{2} \)
83 \( 1 + 2735332 p T + p^{9} T^{2} \)
89 \( 1 - 72141594 T + p^{9} T^{2} \)
97 \( 1 - 228776546 T + p^{9} T^{2} \)
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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

−19.38588205464057563373118747132, −17.28412154973404202505291721511, −16.05597356337634205029049745914, −14.55994146943104185203098315520, −12.27657017195244364929631014671, −11.23201678037774248394382591424, −8.779712859171732142921205524257, −6.64640886152429687539251930687, −3.90725001931207837486535107703, 0, 3.90725001931207837486535107703, 6.64640886152429687539251930687, 8.779712859171732142921205524257, 11.23201678037774248394382591424, 12.27657017195244364929631014671, 14.55994146943104185203098315520, 16.05597356337634205029049745914, 17.28412154973404202505291721511, 19.38588205464057563373118747132

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