Properties

Label 2-4012-1.1-c1-0-0
Degree $2$
Conductor $4012$
Sign $1$
Analytic cond. $32.0359$
Root an. cond. $5.66003$
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  + 0.765·3-s − 3.22·5-s − 4.55·7-s − 2.41·9-s − 3.45·11-s − 1.86·13-s − 2.46·15-s + 17-s − 6.05·19-s − 3.48·21-s + 3.04·23-s + 5.39·25-s − 4.14·27-s − 4.74·29-s + 2.29·31-s − 2.64·33-s + 14.6·35-s + 0.243·37-s − 1.42·39-s − 7.57·41-s − 11.0·43-s + 7.78·45-s − 0.420·47-s + 13.7·49-s + 0.765·51-s − 11.6·53-s + 11.1·55-s + ⋯
L(s)  = 1  + 0.441·3-s − 1.44·5-s − 1.72·7-s − 0.804·9-s − 1.04·11-s − 0.517·13-s − 0.637·15-s + 0.242·17-s − 1.38·19-s − 0.761·21-s + 0.634·23-s + 1.07·25-s − 0.797·27-s − 0.881·29-s + 0.412·31-s − 0.459·33-s + 2.48·35-s + 0.0399·37-s − 0.228·39-s − 1.18·41-s − 1.68·43-s + 1.16·45-s − 0.0613·47-s + 1.96·49-s + 0.107·51-s − 1.60·53-s + 1.49·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $1$
Analytic conductor: \(32.0359\)
Root analytic conductor: \(5.66003\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4012,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09872779870\)
\(L(\frac12)\) \(\approx\) \(0.09872779870\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 - T \)
59 \( 1 - T \)
good3 \( 1 - 0.765T + 3T^{2} \)
5 \( 1 + 3.22T + 5T^{2} \)
7 \( 1 + 4.55T + 7T^{2} \)
11 \( 1 + 3.45T + 11T^{2} \)
13 \( 1 + 1.86T + 13T^{2} \)
19 \( 1 + 6.05T + 19T^{2} \)
23 \( 1 - 3.04T + 23T^{2} \)
29 \( 1 + 4.74T + 29T^{2} \)
31 \( 1 - 2.29T + 31T^{2} \)
37 \( 1 - 0.243T + 37T^{2} \)
41 \( 1 + 7.57T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + 0.420T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
61 \( 1 + 12.8T + 61T^{2} \)
67 \( 1 - 4.15T + 67T^{2} \)
71 \( 1 - 7.72T + 71T^{2} \)
73 \( 1 - 8.51T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 + 1.34T + 83T^{2} \)
89 \( 1 - 4.23T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−8.174942237099686629410974159551, −7.998318880082889330952633722717, −6.96223226694674243516759858205, −6.45976795091417925423887690343, −5.44030756751384007840557480437, −4.56653014956626599578873245841, −3.46717706308028520699670753184, −3.24109800435173310989850069395, −2.30956462548974233192210901900, −0.15971283590652941051328433946, 0.15971283590652941051328433946, 2.30956462548974233192210901900, 3.24109800435173310989850069395, 3.46717706308028520699670753184, 4.56653014956626599578873245841, 5.44030756751384007840557480437, 6.45976795091417925423887690343, 6.96223226694674243516759858205, 7.998318880082889330952633722717, 8.174942237099686629410974159551

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