Properties

Label 2-6008-1.1-c1-0-172
Degree $2$
Conductor $6008$
Sign $-1$
Analytic cond. $47.9741$
Root an. cond. $6.92633$
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  − 0.181·3-s + 3.75·5-s + 4.35·7-s − 2.96·9-s − 3.71·11-s − 5.73·13-s − 0.682·15-s + 0.137·17-s − 1.56·19-s − 0.792·21-s − 5.86·23-s + 9.09·25-s + 1.08·27-s − 7.95·29-s − 7.97·31-s + 0.674·33-s + 16.3·35-s + 3.70·37-s + 1.04·39-s + 2.25·41-s + 9.65·43-s − 11.1·45-s − 9.13·47-s + 12.0·49-s − 0.0250·51-s − 9.27·53-s − 13.9·55-s + ⋯
L(s)  = 1  − 0.104·3-s + 1.67·5-s + 1.64·7-s − 0.988·9-s − 1.11·11-s − 1.58·13-s − 0.176·15-s + 0.0334·17-s − 0.358·19-s − 0.172·21-s − 1.22·23-s + 1.81·25-s + 0.208·27-s − 1.47·29-s − 1.43·31-s + 0.117·33-s + 2.76·35-s + 0.609·37-s + 0.166·39-s + 0.351·41-s + 1.47·43-s − 1.66·45-s − 1.33·47-s + 1.71·49-s − 0.00350·51-s − 1.27·53-s − 1.87·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-1$
Analytic conductor: \(47.9741\)
Root analytic conductor: \(6.92633\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6008,\ (\ :1/2),\ -1)\)

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 \( 1 \)
751 \( 1 - T \)
good3 \( 1 + 0.181T + 3T^{2} \)
5 \( 1 - 3.75T + 5T^{2} \)
7 \( 1 - 4.35T + 7T^{2} \)
11 \( 1 + 3.71T + 11T^{2} \)
13 \( 1 + 5.73T + 13T^{2} \)
17 \( 1 - 0.137T + 17T^{2} \)
19 \( 1 + 1.56T + 19T^{2} \)
23 \( 1 + 5.86T + 23T^{2} \)
29 \( 1 + 7.95T + 29T^{2} \)
31 \( 1 + 7.97T + 31T^{2} \)
37 \( 1 - 3.70T + 37T^{2} \)
41 \( 1 - 2.25T + 41T^{2} \)
43 \( 1 - 9.65T + 43T^{2} \)
47 \( 1 + 9.13T + 47T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 + 5.77T + 61T^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + 1.47T + 83T^{2} \)
89 \( 1 + 2.87T + 89T^{2} \)
97 \( 1 + 5.91T + 97T^{2} \)
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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

−7.76821513949170623400951369322, −7.15843971147650157719385136587, −5.90564548525509064844851264652, −5.59511773825332777970823396648, −5.08544125471823467498953269019, −4.33793740095179562825466144939, −2.82177321360083681860552890711, −2.18617644055366708785987032740, −1.71025339111310228325686296651, 0, 1.71025339111310228325686296651, 2.18617644055366708785987032740, 2.82177321360083681860552890711, 4.33793740095179562825466144939, 5.08544125471823467498953269019, 5.59511773825332777970823396648, 5.90564548525509064844851264652, 7.15843971147650157719385136587, 7.76821513949170623400951369322

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