Properties

Label 2-4012-1.1-c1-0-8
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  − 1.96·3-s − 1.93·5-s − 2.26·7-s + 0.878·9-s + 4.98·11-s + 1.24·13-s + 3.81·15-s + 17-s − 2.47·19-s + 4.46·21-s − 3.49·23-s − 1.24·25-s + 4.17·27-s − 5.88·29-s − 5.18·31-s − 9.82·33-s + 4.39·35-s + 2.85·37-s − 2.45·39-s + 1.07·41-s + 5.36·43-s − 1.70·45-s − 7.66·47-s − 1.85·49-s − 1.96·51-s − 8.25·53-s − 9.66·55-s + ⋯
L(s)  = 1  − 1.13·3-s − 0.866·5-s − 0.857·7-s + 0.292·9-s + 1.50·11-s + 0.345·13-s + 0.985·15-s + 0.242·17-s − 0.568·19-s + 0.974·21-s − 0.729·23-s − 0.249·25-s + 0.803·27-s − 1.09·29-s − 0.931·31-s − 1.71·33-s + 0.742·35-s + 0.470·37-s − 0.392·39-s + 0.167·41-s + 0.817·43-s − 0.253·45-s − 1.11·47-s − 0.265·49-s − 0.275·51-s − 1.13·53-s − 1.30·55-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

\(L(1)\) \(\approx\) \(0.5547613861\)
\(L(\frac12)\) \(\approx\) \(0.5547613861\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.96T + 3T^{2} \)
5 \( 1 + 1.93T + 5T^{2} \)
7 \( 1 + 2.26T + 7T^{2} \)
11 \( 1 - 4.98T + 11T^{2} \)
13 \( 1 - 1.24T + 13T^{2} \)
19 \( 1 + 2.47T + 19T^{2} \)
23 \( 1 + 3.49T + 23T^{2} \)
29 \( 1 + 5.88T + 29T^{2} \)
31 \( 1 + 5.18T + 31T^{2} \)
37 \( 1 - 2.85T + 37T^{2} \)
41 \( 1 - 1.07T + 41T^{2} \)
43 \( 1 - 5.36T + 43T^{2} \)
47 \( 1 + 7.66T + 47T^{2} \)
53 \( 1 + 8.25T + 53T^{2} \)
61 \( 1 - 0.588T + 61T^{2} \)
67 \( 1 + 7.16T + 67T^{2} \)
71 \( 1 + 2.36T + 71T^{2} \)
73 \( 1 - 5.38T + 73T^{2} \)
79 \( 1 + 6.34T + 79T^{2} \)
83 \( 1 - 6.86T + 83T^{2} \)
89 \( 1 - 3.68T + 89T^{2} \)
97 \( 1 - 0.642T + 97T^{2} \)
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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

−8.436450423851305536284621583775, −7.58431632217960879555367179934, −6.80459233652649489975952330978, −6.17475897594283514799713030261, −5.74793261308722672870969761390, −4.58253783205464014801799255573, −3.89764634573792434587692812136, −3.26980906821309760704647242748, −1.72447263357124791481359615245, −0.45126733055553191514823990335, 0.45126733055553191514823990335, 1.72447263357124791481359615245, 3.26980906821309760704647242748, 3.89764634573792434587692812136, 4.58253783205464014801799255573, 5.74793261308722672870969761390, 6.17475897594283514799713030261, 6.80459233652649489975952330978, 7.58431632217960879555367179934, 8.436450423851305536284621583775

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