Properties

Label 2-4014-1.1-c1-0-16
Degree $2$
Conductor $4014$
Sign $1$
Analytic cond. $32.0519$
Root an. cond. $5.66144$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4.32·5-s + 2.79·7-s − 8-s + 4.32·10-s + 2.77·11-s − 4.55·13-s − 2.79·14-s + 16-s + 7.65·17-s − 0.819·19-s − 4.32·20-s − 2.77·22-s + 3.06·23-s + 13.7·25-s + 4.55·26-s + 2.79·28-s − 2.92·29-s − 1.60·31-s − 32-s − 7.65·34-s − 12.1·35-s − 4.99·37-s + 0.819·38-s + 4.32·40-s − 4.19·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.93·5-s + 1.05·7-s − 0.353·8-s + 1.36·10-s + 0.835·11-s − 1.26·13-s − 0.747·14-s + 0.250·16-s + 1.85·17-s − 0.187·19-s − 0.967·20-s − 0.590·22-s + 0.638·23-s + 2.74·25-s + 0.892·26-s + 0.528·28-s − 0.543·29-s − 0.287·31-s − 0.176·32-s − 1.31·34-s − 2.04·35-s − 0.820·37-s + 0.132·38-s + 0.684·40-s − 0.654·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4014\)    =    \(2 \cdot 3^{2} \cdot 223\)
Sign: $1$
Analytic conductor: \(32.0519\)
Root analytic conductor: \(5.66144\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4014,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9409066920\)
\(L(\frac12)\) \(\approx\) \(0.9409066920\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
223 \( 1 + T \)
good5 \( 1 + 4.32T + 5T^{2} \)
7 \( 1 - 2.79T + 7T^{2} \)
11 \( 1 - 2.77T + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 + 0.819T + 19T^{2} \)
23 \( 1 - 3.06T + 23T^{2} \)
29 \( 1 + 2.92T + 29T^{2} \)
31 \( 1 + 1.60T + 31T^{2} \)
37 \( 1 + 4.99T + 37T^{2} \)
41 \( 1 + 4.19T + 41T^{2} \)
43 \( 1 - 6.95T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 2.54T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 1.42T + 61T^{2} \)
67 \( 1 - 5.12T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 + 9.56T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 6.23T + 83T^{2} \)
89 \( 1 - 3.15T + 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.268644236883803349553361477456, −7.68646826565534599755193960359, −7.42008239361119295203319582626, −6.59495677384890405657269045299, −5.25496710716203657877551380090, −4.69887302947875826716939948280, −3.72396810734122748735942188883, −3.09485880205996195714066071804, −1.68133347259820368418004461284, −0.63980567483382581177600725725, 0.63980567483382581177600725725, 1.68133347259820368418004461284, 3.09485880205996195714066071804, 3.72396810734122748735942188883, 4.69887302947875826716939948280, 5.25496710716203657877551380090, 6.59495677384890405657269045299, 7.42008239361119295203319582626, 7.68646826565534599755193960359, 8.268644236883803349553361477456

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