Properties

Label 2-4006-1.1-c1-0-74
Degree $2$
Conductor $4006$
Sign $1$
Analytic cond. $31.9880$
Root an. cond. $5.65579$
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 + 0.885·3-s + 4-s − 1.00·5-s + 0.885·6-s + 2.56·7-s + 8-s − 2.21·9-s − 1.00·10-s + 2.77·11-s + 0.885·12-s + 5.85·13-s + 2.56·14-s − 0.885·15-s + 16-s + 2.27·17-s − 2.21·18-s − 0.846·19-s − 1.00·20-s + 2.26·21-s + 2.77·22-s + 0.431·23-s + 0.885·24-s − 3.99·25-s + 5.85·26-s − 4.61·27-s + 2.56·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.511·3-s + 0.5·4-s − 0.447·5-s + 0.361·6-s + 0.967·7-s + 0.353·8-s − 0.738·9-s − 0.316·10-s + 0.837·11-s + 0.255·12-s + 1.62·13-s + 0.684·14-s − 0.228·15-s + 0.250·16-s + 0.551·17-s − 0.522·18-s − 0.194·19-s − 0.223·20-s + 0.494·21-s + 0.592·22-s + 0.0899·23-s + 0.180·24-s − 0.799·25-s + 1.14·26-s − 0.888·27-s + 0.483·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4006\)    =    \(2 \cdot 2003\)
Sign: $1$
Analytic conductor: \(31.9880\)
Root analytic conductor: \(5.65579\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4006,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.154718092\)
\(L(\frac12)\) \(\approx\) \(4.154718092\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
2003 \( 1 + T \)
good3 \( 1 - 0.885T + 3T^{2} \)
5 \( 1 + 1.00T + 5T^{2} \)
7 \( 1 - 2.56T + 7T^{2} \)
11 \( 1 - 2.77T + 11T^{2} \)
13 \( 1 - 5.85T + 13T^{2} \)
17 \( 1 - 2.27T + 17T^{2} \)
19 \( 1 + 0.846T + 19T^{2} \)
23 \( 1 - 0.431T + 23T^{2} \)
29 \( 1 - 5.20T + 29T^{2} \)
31 \( 1 - 2.98T + 31T^{2} \)
37 \( 1 - 0.383T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 4.57T + 43T^{2} \)
47 \( 1 + 4.08T + 47T^{2} \)
53 \( 1 - 2.12T + 53T^{2} \)
59 \( 1 - 9.39T + 59T^{2} \)
61 \( 1 - 1.84T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 6.55T + 73T^{2} \)
79 \( 1 + 6.54T + 79T^{2} \)
83 \( 1 - 1.16T + 83T^{2} \)
89 \( 1 - 8.19T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−8.299329424318420898461358031934, −7.962207029694079966706731006678, −6.87360410625374846676465655240, −6.16316023857386415474634602139, −5.45736580895130526989458054384, −4.54261043230882256311185539920, −3.75990987559138110569625588540, −3.24370280781482558890928196045, −2.06887251272903720811522971195, −1.11865637425401466323895845354, 1.11865637425401466323895845354, 2.06887251272903720811522971195, 3.24370280781482558890928196045, 3.75990987559138110569625588540, 4.54261043230882256311185539920, 5.45736580895130526989458054384, 6.16316023857386415474634602139, 6.87360410625374846676465655240, 7.962207029694079966706731006678, 8.299329424318420898461358031934

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