Properties

Label 2-4034-1.1-c1-0-71
Degree $2$
Conductor $4034$
Sign $-1$
Analytic cond. $32.2116$
Root an. cond. $5.67553$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.39·3-s + 4-s − 1.11·5-s + 2.39·6-s + 2.17·7-s − 8-s + 2.75·9-s + 1.11·10-s − 3.08·11-s − 2.39·12-s + 2.44·13-s − 2.17·14-s + 2.67·15-s + 16-s + 4.89·17-s − 2.75·18-s − 4.74·19-s − 1.11·20-s − 5.22·21-s + 3.08·22-s − 3.76·23-s + 2.39·24-s − 3.75·25-s − 2.44·26-s + 0.577·27-s + 2.17·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.38·3-s + 0.5·4-s − 0.498·5-s + 0.979·6-s + 0.822·7-s − 0.353·8-s + 0.919·9-s + 0.352·10-s − 0.930·11-s − 0.692·12-s + 0.676·13-s − 0.581·14-s + 0.691·15-s + 0.250·16-s + 1.18·17-s − 0.650·18-s − 1.08·19-s − 0.249·20-s − 1.13·21-s + 0.657·22-s − 0.785·23-s + 0.489·24-s − 0.751·25-s − 0.478·26-s + 0.111·27-s + 0.411·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4034\)    =    \(2 \cdot 2017\)
Sign: $-1$
Analytic conductor: \(32.2116\)
Root analytic conductor: \(5.67553\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4034,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
2017 \( 1 + T \)
good3 \( 1 + 2.39T + 3T^{2} \)
5 \( 1 + 1.11T + 5T^{2} \)
7 \( 1 - 2.17T + 7T^{2} \)
11 \( 1 + 3.08T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 + 4.74T + 19T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 + 7.37T + 31T^{2} \)
37 \( 1 - 3.03T + 37T^{2} \)
41 \( 1 - 9.67T + 41T^{2} \)
43 \( 1 + 6.41T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 9.19T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 1.55T + 67T^{2} \)
71 \( 1 - 7.01T + 71T^{2} \)
73 \( 1 + 6.82T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 - 9.20T + 83T^{2} \)
89 \( 1 - 5.85T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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

−8.194204723091146079097518306789, −7.46863294700698415142006312085, −6.51925108991839034294650185518, −5.93269009335895467067559823702, −5.20769151481827045620304583914, −4.48808078088130808423560738497, −3.44653091034419765074445381161, −2.14361578619918769128470031449, −1.05527867445993174055077692482, 0, 1.05527867445993174055077692482, 2.14361578619918769128470031449, 3.44653091034419765074445381161, 4.48808078088130808423560738497, 5.20769151481827045620304583914, 5.93269009335895467067559823702, 6.51925108991839034294650185518, 7.46863294700698415142006312085, 8.194204723091146079097518306789

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