Properties

Label 2-8046-1.1-c1-0-130
Degree $2$
Conductor $8046$
Sign $1$
Analytic cond. $64.2476$
Root an. cond. $8.01546$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 1.33·5-s + 2.26·7-s + 8-s + 1.33·10-s + 5.80·11-s + 2.07·13-s + 2.26·14-s + 16-s − 5.11·17-s + 8.67·19-s + 1.33·20-s + 5.80·22-s + 3.21·23-s − 3.21·25-s + 2.07·26-s + 2.26·28-s + 3.55·29-s − 4.87·31-s + 32-s − 5.11·34-s + 3.02·35-s + 4.63·37-s + 8.67·38-s + 1.33·40-s + 8.09·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.597·5-s + 0.856·7-s + 0.353·8-s + 0.422·10-s + 1.75·11-s + 0.576·13-s + 0.605·14-s + 0.250·16-s − 1.24·17-s + 1.98·19-s + 0.298·20-s + 1.23·22-s + 0.670·23-s − 0.643·25-s + 0.407·26-s + 0.428·28-s + 0.659·29-s − 0.876·31-s + 0.176·32-s − 0.877·34-s + 0.511·35-s + 0.762·37-s + 1.40·38-s + 0.211·40-s + 1.26·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8046\)    =    \(2 \cdot 3^{3} \cdot 149\)
Sign: $1$
Analytic conductor: \(64.2476\)
Root analytic conductor: \(8.01546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8046,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.272080390\)
\(L(\frac12)\) \(\approx\) \(5.272080390\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
149 \( 1 + T \)
good5 \( 1 - 1.33T + 5T^{2} \)
7 \( 1 - 2.26T + 7T^{2} \)
11 \( 1 - 5.80T + 11T^{2} \)
13 \( 1 - 2.07T + 13T^{2} \)
17 \( 1 + 5.11T + 17T^{2} \)
19 \( 1 - 8.67T + 19T^{2} \)
23 \( 1 - 3.21T + 23T^{2} \)
29 \( 1 - 3.55T + 29T^{2} \)
31 \( 1 + 4.87T + 31T^{2} \)
37 \( 1 - 4.63T + 37T^{2} \)
41 \( 1 - 8.09T + 41T^{2} \)
43 \( 1 + 7.14T + 43T^{2} \)
47 \( 1 + 4.49T + 47T^{2} \)
53 \( 1 - 1.64T + 53T^{2} \)
59 \( 1 - 1.45T + 59T^{2} \)
61 \( 1 - 2.95T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + 6.47T + 71T^{2} \)
73 \( 1 + 3.65T + 73T^{2} \)
79 \( 1 - 4.22T + 79T^{2} \)
83 \( 1 - 0.157T + 83T^{2} \)
89 \( 1 + 4.44T + 89T^{2} \)
97 \( 1 - 8.30T + 97T^{2} \)
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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

−7.68217146072066243318564570928, −6.96745538508467451044362957502, −6.37500335141023178146462794779, −5.73234810834127307741665947428, −5.00757640536969134098524020553, −4.31002096618385215586264516023, −3.64423034956276284151077944135, −2.74821543636996184636843656164, −1.66922891221371481565432826486, −1.19050477223277177421934249060, 1.19050477223277177421934249060, 1.66922891221371481565432826486, 2.74821543636996184636843656164, 3.64423034956276284151077944135, 4.31002096618385215586264516023, 5.00757640536969134098524020553, 5.73234810834127307741665947428, 6.37500335141023178146462794779, 6.96745538508467451044362957502, 7.68217146072066243318564570928

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