Properties

Label 2-9408-1.1-c1-0-147
Degree $2$
Conductor $9408$
Sign $-1$
Analytic cond. $75.1232$
Root an. cond. $8.66736$
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 + 3.04·5-s + 9-s − 3.93·11-s + 4.88·13-s − 3.04·15-s + 5.34·17-s + 2.30·19-s − 7.93·23-s + 4.25·25-s − 27-s − 5.55·29-s − 0.645·31-s + 3.93·33-s − 5.65·37-s − 4.88·39-s − 10.0·41-s − 8.91·43-s + 3.04·45-s − 6.61·47-s − 5.34·51-s − 1.25·53-s − 11.9·55-s − 2.30·57-s − 3.04·59-s − 2.97·61-s + 14.8·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.36·5-s + 0.333·9-s − 1.18·11-s + 1.35·13-s − 0.785·15-s + 1.29·17-s + 0.528·19-s − 1.65·23-s + 0.851·25-s − 0.192·27-s − 1.03·29-s − 0.115·31-s + 0.684·33-s − 0.929·37-s − 0.782·39-s − 1.57·41-s − 1.35·43-s + 0.453·45-s − 0.964·47-s − 0.748·51-s − 0.172·53-s − 1.61·55-s − 0.304·57-s − 0.396·59-s − 0.380·61-s + 1.84·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(75.1232\)
Root analytic conductor: \(8.66736\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 3.04T + 5T^{2} \)
11 \( 1 + 3.93T + 11T^{2} \)
13 \( 1 - 4.88T + 13T^{2} \)
17 \( 1 - 5.34T + 17T^{2} \)
19 \( 1 - 2.30T + 19T^{2} \)
23 \( 1 + 7.93T + 23T^{2} \)
29 \( 1 + 5.55T + 29T^{2} \)
31 \( 1 + 0.645T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 + 8.91T + 43T^{2} \)
47 \( 1 + 6.61T + 47T^{2} \)
53 \( 1 + 1.25T + 53T^{2} \)
59 \( 1 + 3.04T + 59T^{2} \)
61 \( 1 + 2.97T + 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + 4.67T + 73T^{2} \)
79 \( 1 + 1.05T + 79T^{2} \)
83 \( 1 + 8.60T + 83T^{2} \)
89 \( 1 + 4.85T + 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−7.29629830116614954041005041749, −6.47774639318259680214397815119, −5.79006508398158899589196735747, −5.55111333698586453285324416415, −4.88834880195464557095852231511, −3.71039628285252853138341468707, −3.09182820842358978527202303002, −1.88348477703646407683202705256, −1.45349379591543604287854601545, 0, 1.45349379591543604287854601545, 1.88348477703646407683202705256, 3.09182820842358978527202303002, 3.71039628285252853138341468707, 4.88834880195464557095852231511, 5.55111333698586453285324416415, 5.79006508398158899589196735747, 6.47774639318259680214397815119, 7.29629830116614954041005041749

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