Properties

Label 2-4016-1.1-c1-0-112
Degree $2$
Conductor $4016$
Sign $-1$
Analytic cond. $32.0679$
Root an. cond. $5.66285$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·3-s − 1.89·5-s − 0.462·7-s + 2.81·9-s + 0.172·11-s + 0.314·13-s − 4.56·15-s − 7.25·17-s − 0.194·19-s − 1.11·21-s − 0.825·23-s − 1.41·25-s − 0.440·27-s + 3.71·29-s + 7.97·31-s + 0.415·33-s + 0.876·35-s − 6.20·37-s + 0.757·39-s − 1.31·41-s − 3.94·43-s − 5.33·45-s + 2.22·47-s − 6.78·49-s − 17.4·51-s + 10.0·53-s − 0.325·55-s + ⋯
L(s)  = 1  + 1.39·3-s − 0.846·5-s − 0.174·7-s + 0.939·9-s + 0.0518·11-s + 0.0871·13-s − 1.17·15-s − 1.75·17-s − 0.0445·19-s − 0.243·21-s − 0.172·23-s − 0.282·25-s − 0.0847·27-s + 0.689·29-s + 1.43·31-s + 0.0722·33-s + 0.148·35-s − 1.02·37-s + 0.121·39-s − 0.205·41-s − 0.601·43-s − 0.795·45-s + 0.324·47-s − 0.969·49-s − 2.44·51-s + 1.38·53-s − 0.0439·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4016\)    =    \(2^{4} \cdot 251\)
Sign: $-1$
Analytic conductor: \(32.0679\)
Root analytic conductor: \(5.66285\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4016,\ (\ :1/2),\ -1)\)

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 \( 1 \)
251 \( 1 + T \)
good3 \( 1 - 2.41T + 3T^{2} \)
5 \( 1 + 1.89T + 5T^{2} \)
7 \( 1 + 0.462T + 7T^{2} \)
11 \( 1 - 0.172T + 11T^{2} \)
13 \( 1 - 0.314T + 13T^{2} \)
17 \( 1 + 7.25T + 17T^{2} \)
19 \( 1 + 0.194T + 19T^{2} \)
23 \( 1 + 0.825T + 23T^{2} \)
29 \( 1 - 3.71T + 29T^{2} \)
31 \( 1 - 7.97T + 31T^{2} \)
37 \( 1 + 6.20T + 37T^{2} \)
41 \( 1 + 1.31T + 41T^{2} \)
43 \( 1 + 3.94T + 43T^{2} \)
47 \( 1 - 2.22T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + 6.02T + 59T^{2} \)
61 \( 1 + 9.74T + 61T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 + 9.68T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 - 9.75T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 - 8.75T + 89T^{2} \)
97 \( 1 - 9.75T + 97T^{2} \)
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

−8.236745317357673416576472895395, −7.50948440804257496337662282983, −6.82839626322867221479796295085, −6.03317835605111554792149020505, −4.66018761253850194188973969474, −4.19705566364519988660733895348, −3.30212626280635608150194630592, −2.65595391378646636224193591514, −1.68219006718632468243267875047, 0, 1.68219006718632468243267875047, 2.65595391378646636224193591514, 3.30212626280635608150194630592, 4.19705566364519988660733895348, 4.66018761253850194188973969474, 6.03317835605111554792149020505, 6.82839626322867221479796295085, 7.50948440804257496337662282983, 8.236745317357673416576472895395

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