Properties

Label 2-4001-1.1-c1-0-126
Degree $2$
Conductor $4001$
Sign $1$
Analytic cond. $31.9481$
Root an. cond. $5.65226$
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  + 1.92·2-s + 1.55·3-s + 1.70·4-s − 3.80·5-s + 3.00·6-s − 0.241·7-s − 0.560·8-s − 0.568·9-s − 7.32·10-s + 4.51·11-s + 2.66·12-s − 2.64·13-s − 0.465·14-s − 5.93·15-s − 4.49·16-s − 0.0403·17-s − 1.09·18-s + 7.82·19-s − 6.50·20-s − 0.376·21-s + 8.69·22-s + 9.16·23-s − 0.873·24-s + 9.48·25-s − 5.08·26-s − 5.56·27-s − 0.412·28-s + ⋯
L(s)  = 1  + 1.36·2-s + 0.900·3-s + 0.854·4-s − 1.70·5-s + 1.22·6-s − 0.0912·7-s − 0.198·8-s − 0.189·9-s − 2.31·10-s + 1.36·11-s + 0.769·12-s − 0.732·13-s − 0.124·14-s − 1.53·15-s − 1.12·16-s − 0.00978·17-s − 0.258·18-s + 1.79·19-s − 1.45·20-s − 0.0821·21-s + 1.85·22-s + 1.91·23-s − 0.178·24-s + 1.89·25-s − 0.997·26-s − 1.07·27-s − 0.0780·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4001\)
Sign: $1$
Analytic conductor: \(31.9481\)
Root analytic conductor: \(5.65226\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.861965194\)
\(L(\frac12)\) \(\approx\) \(3.861965194\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1+O(T) \)
good2 \( 1 - 1.92T + 2T^{2} \)
3 \( 1 - 1.55T + 3T^{2} \)
5 \( 1 + 3.80T + 5T^{2} \)
7 \( 1 + 0.241T + 7T^{2} \)
11 \( 1 - 4.51T + 11T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 + 0.0403T + 17T^{2} \)
19 \( 1 - 7.82T + 19T^{2} \)
23 \( 1 - 9.16T + 23T^{2} \)
29 \( 1 - 4.49T + 29T^{2} \)
31 \( 1 - 7.37T + 31T^{2} \)
37 \( 1 - 4.70T + 37T^{2} \)
41 \( 1 - 0.360T + 41T^{2} \)
43 \( 1 + 8.36T + 43T^{2} \)
47 \( 1 - 9.02T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 2.84T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 4.86T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 - 2.70T + 83T^{2} \)
89 \( 1 - 17.9T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−8.266614435252755300680879136796, −7.69571620169918866323529687122, −6.92246564706337895331281543561, −6.31348952143009589233938347544, −4.92656035341853725938897797612, −4.73038055234540796074487979801, −3.56793562480512179654564873355, −3.37561433550679814122936246499, −2.65263379300532954714121177520, −0.890426521054675016530915629882, 0.890426521054675016530915629882, 2.65263379300532954714121177520, 3.37561433550679814122936246499, 3.56793562480512179654564873355, 4.73038055234540796074487979801, 4.92656035341853725938897797612, 6.31348952143009589233938347544, 6.92246564706337895331281543561, 7.69571620169918866323529687122, 8.266614435252755300680879136796

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