Properties

Label 2-8046-1.1-c1-0-114
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 3.58·5-s − 4.00·7-s + 8-s − 3.58·10-s + 1.14·11-s + 0.738·13-s − 4.00·14-s + 16-s + 4.19·17-s + 1.58·19-s − 3.58·20-s + 1.14·22-s − 2.22·23-s + 7.84·25-s + 0.738·26-s − 4.00·28-s + 2.40·29-s + 4.76·31-s + 32-s + 4.19·34-s + 14.3·35-s − 9.04·37-s + 1.58·38-s − 3.58·40-s + 3.19·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.60·5-s − 1.51·7-s + 0.353·8-s − 1.13·10-s + 0.346·11-s + 0.204·13-s − 1.06·14-s + 0.250·16-s + 1.01·17-s + 0.364·19-s − 0.801·20-s + 0.244·22-s − 0.463·23-s + 1.56·25-s + 0.144·26-s − 0.756·28-s + 0.446·29-s + 0.855·31-s + 0.176·32-s + 0.719·34-s + 2.42·35-s − 1.48·37-s + 0.257·38-s − 0.566·40-s + 0.498·41-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: \(1\)
Selberg data: \((2,\ 8046,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
149 \( 1 - T \)
good5 \( 1 + 3.58T + 5T^{2} \)
7 \( 1 + 4.00T + 7T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
13 \( 1 - 0.738T + 13T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
19 \( 1 - 1.58T + 19T^{2} \)
23 \( 1 + 2.22T + 23T^{2} \)
29 \( 1 - 2.40T + 29T^{2} \)
31 \( 1 - 4.76T + 31T^{2} \)
37 \( 1 + 9.04T + 37T^{2} \)
41 \( 1 - 3.19T + 41T^{2} \)
43 \( 1 - 5.30T + 43T^{2} \)
47 \( 1 + 5.72T + 47T^{2} \)
53 \( 1 + 0.714T + 53T^{2} \)
59 \( 1 - 6.45T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 - 6.34T + 67T^{2} \)
71 \( 1 + 4.19T + 71T^{2} \)
73 \( 1 - 7.45T + 73T^{2} \)
79 \( 1 + 2.56T + 79T^{2} \)
83 \( 1 + 9.66T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 5.33T + 97T^{2} \)
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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

−7.41159308882243041536041525757, −6.73135628544578251345864248877, −6.20507461899691758047892929455, −5.35423383541559845351881547734, −4.47148781429285582959868834190, −3.72935469909954844465906017686, −3.38938134322790743823689423077, −2.68167142082102771160215082940, −1.12277599167935978311161255993, 0, 1.12277599167935978311161255993, 2.68167142082102771160215082940, 3.38938134322790743823689423077, 3.72935469909954844465906017686, 4.47148781429285582959868834190, 5.35423383541559845351881547734, 6.20507461899691758047892929455, 6.73135628544578251345864248877, 7.41159308882243041536041525757

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