Properties

Label 2-4016-1.1-c1-0-2
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.00508·3-s − 3.22·5-s − 4.03·7-s − 2.99·9-s + 0.837·11-s + 2.15·13-s − 0.0164·15-s − 5.70·17-s − 3.20·19-s − 0.0205·21-s − 7.31·23-s + 5.40·25-s − 0.0305·27-s − 10.2·29-s − 2.92·31-s + 0.00426·33-s + 13.0·35-s + 1.03·37-s + 0.0109·39-s − 6.57·41-s + 8.92·43-s + 9.67·45-s + 0.200·47-s + 9.30·49-s − 0.0290·51-s − 4.20·53-s − 2.70·55-s + ⋯
L(s)  = 1  + 0.00293·3-s − 1.44·5-s − 1.52·7-s − 0.999·9-s + 0.252·11-s + 0.597·13-s − 0.00423·15-s − 1.38·17-s − 0.735·19-s − 0.00448·21-s − 1.52·23-s + 1.08·25-s − 0.00587·27-s − 1.89·29-s − 0.525·31-s + 0.000741·33-s + 2.20·35-s + 0.170·37-s + 0.00175·39-s − 1.02·41-s + 1.36·43-s + 1.44·45-s + 0.0292·47-s + 1.32·49-s − 0.00406·51-s − 0.577·53-s − 0.364·55-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: \(0\)
Selberg data: \((2,\ 4016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09893459332\)
\(L(\frac12)\) \(\approx\) \(0.09893459332\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.00508T + 3T^{2} \)
5 \( 1 + 3.22T + 5T^{2} \)
7 \( 1 + 4.03T + 7T^{2} \)
11 \( 1 - 0.837T + 11T^{2} \)
13 \( 1 - 2.15T + 13T^{2} \)
17 \( 1 + 5.70T + 17T^{2} \)
19 \( 1 + 3.20T + 19T^{2} \)
23 \( 1 + 7.31T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + 2.92T + 31T^{2} \)
37 \( 1 - 1.03T + 37T^{2} \)
41 \( 1 + 6.57T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 - 0.200T + 47T^{2} \)
53 \( 1 + 4.20T + 53T^{2} \)
59 \( 1 - 4.19T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 5.06T + 67T^{2} \)
71 \( 1 - 6.00T + 71T^{2} \)
73 \( 1 + 3.36T + 73T^{2} \)
79 \( 1 + 6.23T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + 7.88T + 89T^{2} \)
97 \( 1 - 4.91T + 97T^{2} \)
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

−8.519077399438792171796073793942, −7.73515345260515228600546736569, −6.97125330448816722976523692667, −6.24764857475565236905640541014, −5.71238465185635171779664628973, −4.32803781282621288337944353056, −3.81681942806280160124919532926, −3.19444218046377932613847460665, −2.12932585250450049667872892084, −0.16582806354601031248151206204, 0.16582806354601031248151206204, 2.12932585250450049667872892084, 3.19444218046377932613847460665, 3.81681942806280160124919532926, 4.32803781282621288337944353056, 5.71238465185635171779664628973, 6.24764857475565236905640541014, 6.97125330448816722976523692667, 7.73515345260515228600546736569, 8.519077399438792171796073793942

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