Properties

Label 2-8001-1.1-c1-0-116
Degree $2$
Conductor $8001$
Sign $1$
Analytic cond. $63.8883$
Root an. cond. $7.99301$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.532·2-s − 1.71·4-s + 0.118·5-s − 7-s + 1.97·8-s − 0.0629·10-s + 5.22·11-s − 2.19·13-s + 0.532·14-s + 2.37·16-s + 1.28·17-s + 8.34·19-s − 0.202·20-s − 2.78·22-s + 9.05·23-s − 4.98·25-s + 1.16·26-s + 1.71·28-s + 9.26·29-s + 6.76·31-s − 5.22·32-s − 0.682·34-s − 0.118·35-s + 6.01·37-s − 4.44·38-s + 0.233·40-s + 4.32·41-s + ⋯
L(s)  = 1  − 0.376·2-s − 0.858·4-s + 0.0528·5-s − 0.377·7-s + 0.699·8-s − 0.0198·10-s + 1.57·11-s − 0.607·13-s + 0.142·14-s + 0.594·16-s + 0.310·17-s + 1.91·19-s − 0.0453·20-s − 0.593·22-s + 1.88·23-s − 0.997·25-s + 0.228·26-s + 0.324·28-s + 1.72·29-s + 1.21·31-s − 0.923·32-s − 0.117·34-s − 0.0199·35-s + 0.989·37-s − 0.720·38-s + 0.0369·40-s + 0.675·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
Sign: $1$
Analytic conductor: \(63.8883\)
Root analytic conductor: \(7.99301\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.726304133\)
\(L(\frac12)\) \(\approx\) \(1.726304133\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
127 \( 1 + T \)
good2 \( 1 + 0.532T + 2T^{2} \)
5 \( 1 - 0.118T + 5T^{2} \)
11 \( 1 - 5.22T + 11T^{2} \)
13 \( 1 + 2.19T + 13T^{2} \)
17 \( 1 - 1.28T + 17T^{2} \)
19 \( 1 - 8.34T + 19T^{2} \)
23 \( 1 - 9.05T + 23T^{2} \)
29 \( 1 - 9.26T + 29T^{2} \)
31 \( 1 - 6.76T + 31T^{2} \)
37 \( 1 - 6.01T + 37T^{2} \)
41 \( 1 - 4.32T + 41T^{2} \)
43 \( 1 - 2.14T + 43T^{2} \)
47 \( 1 + 1.79T + 47T^{2} \)
53 \( 1 + 4.27T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 - 4.12T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 6.41T + 83T^{2} \)
89 \( 1 - 3.69T + 89T^{2} \)
97 \( 1 - 6.10T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.77691059328800700879902220613, −7.32430607188947911998383569054, −6.46475382082623368698939597325, −5.81915523075257178397731282638, −4.80142958918882131220635365050, −4.49218653610256359809160474217, −3.42006399009620672957470394336, −2.88120147327784581202385663148, −1.32016569722918701813235200476, −0.829046382657367290042310738995, 0.829046382657367290042310738995, 1.32016569722918701813235200476, 2.88120147327784581202385663148, 3.42006399009620672957470394336, 4.49218653610256359809160474217, 4.80142958918882131220635365050, 5.81915523075257178397731282638, 6.46475382082623368698939597325, 7.32430607188947911998383569054, 7.77691059328800700879902220613

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