Properties

Label 2-4022-1.1-c1-0-29
Degree $2$
Conductor $4022$
Sign $1$
Analytic cond. $32.1158$
Root an. cond. $5.66708$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.26·3-s + 4-s + 1.05·5-s + 1.26·6-s + 0.0899·7-s − 8-s − 1.38·9-s − 1.05·10-s + 4.74·11-s − 1.26·12-s + 2.51·13-s − 0.0899·14-s − 1.33·15-s + 16-s − 7.81·17-s + 1.38·18-s − 4.19·19-s + 1.05·20-s − 0.114·21-s − 4.74·22-s − 3.87·23-s + 1.26·24-s − 3.89·25-s − 2.51·26-s + 5.57·27-s + 0.0899·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.732·3-s + 0.5·4-s + 0.470·5-s + 0.518·6-s + 0.0340·7-s − 0.353·8-s − 0.463·9-s − 0.332·10-s + 1.42·11-s − 0.366·12-s + 0.696·13-s − 0.0240·14-s − 0.344·15-s + 0.250·16-s − 1.89·17-s + 0.327·18-s − 0.963·19-s + 0.235·20-s − 0.0249·21-s − 1.01·22-s − 0.807·23-s + 0.259·24-s − 0.778·25-s − 0.492·26-s + 1.07·27-s + 0.0170·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4022\)    =    \(2 \cdot 2011\)
Sign: $1$
Analytic conductor: \(32.1158\)
Root analytic conductor: \(5.66708\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4022,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9684829668\)
\(L(\frac12)\) \(\approx\) \(0.9684829668\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
2011 \( 1 - T \)
good3 \( 1 + 1.26T + 3T^{2} \)
5 \( 1 - 1.05T + 5T^{2} \)
7 \( 1 - 0.0899T + 7T^{2} \)
11 \( 1 - 4.74T + 11T^{2} \)
13 \( 1 - 2.51T + 13T^{2} \)
17 \( 1 + 7.81T + 17T^{2} \)
19 \( 1 + 4.19T + 19T^{2} \)
23 \( 1 + 3.87T + 23T^{2} \)
29 \( 1 - 1.30T + 29T^{2} \)
31 \( 1 - 6.51T + 31T^{2} \)
37 \( 1 - 8.50T + 37T^{2} \)
41 \( 1 - 7.45T + 41T^{2} \)
43 \( 1 - 7.56T + 43T^{2} \)
47 \( 1 - 3.50T + 47T^{2} \)
53 \( 1 + 2.44T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 - 4.35T + 67T^{2} \)
71 \( 1 - 16.6T + 71T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 + 9.33T + 79T^{2} \)
83 \( 1 - 3.61T + 83T^{2} \)
89 \( 1 + 4.48T + 89T^{2} \)
97 \( 1 - 7.97T + 97T^{2} \)
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

−8.531860868506985127612883611814, −7.88456801828356869634503949054, −6.65404452411104022818357747566, −6.25436857404000844979427338309, −5.97480809261255239214781024782, −4.58769624199704884423951981403, −4.02390554061708598920675196259, −2.64814736336672900225724567103, −1.80188587774225353127256899351, −0.65004558391671695292358896747, 0.65004558391671695292358896747, 1.80188587774225353127256899351, 2.64814736336672900225724567103, 4.02390554061708598920675196259, 4.58769624199704884423951981403, 5.97480809261255239214781024782, 6.25436857404000844979427338309, 6.65404452411104022818357747566, 7.88456801828356869634503949054, 8.531860868506985127612883611814

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