Properties

Label 2-4030-1.1-c1-0-105
Degree $2$
Conductor $4030$
Sign $-1$
Analytic cond. $32.1797$
Root an. cond. $5.67271$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.642·3-s + 4-s + 5-s − 0.642·6-s + 1.07·7-s + 8-s − 2.58·9-s + 10-s − 6.49·11-s − 0.642·12-s + 13-s + 1.07·14-s − 0.642·15-s + 16-s + 3.21·17-s − 2.58·18-s + 2.94·19-s + 20-s − 0.690·21-s − 6.49·22-s − 6.91·23-s − 0.642·24-s + 25-s + 26-s + 3.58·27-s + 1.07·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.370·3-s + 0.5·4-s + 0.447·5-s − 0.262·6-s + 0.406·7-s + 0.353·8-s − 0.862·9-s + 0.316·10-s − 1.95·11-s − 0.185·12-s + 0.277·13-s + 0.287·14-s − 0.165·15-s + 0.250·16-s + 0.778·17-s − 0.609·18-s + 0.674·19-s + 0.223·20-s − 0.150·21-s − 1.38·22-s − 1.44·23-s − 0.131·24-s + 0.200·25-s + 0.196·26-s + 0.690·27-s + 0.203·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
Sign: $-1$
Analytic conductor: \(32.1797\)
Root analytic conductor: \(5.67271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4030,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 - T \)
31 \( 1 - T \)
good3 \( 1 + 0.642T + 3T^{2} \)
7 \( 1 - 1.07T + 7T^{2} \)
11 \( 1 + 6.49T + 11T^{2} \)
17 \( 1 - 3.21T + 17T^{2} \)
19 \( 1 - 2.94T + 19T^{2} \)
23 \( 1 + 6.91T + 23T^{2} \)
29 \( 1 + 8.00T + 29T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 + 12.5T + 41T^{2} \)
43 \( 1 + 2.23T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 - 2.46T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 1.39T + 61T^{2} \)
67 \( 1 + 8.07T + 67T^{2} \)
71 \( 1 + 6.30T + 71T^{2} \)
73 \( 1 + 7.59T + 73T^{2} \)
79 \( 1 + 6.27T + 79T^{2} \)
83 \( 1 + 2.02T + 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−7.88505706785551977188943828804, −7.48161600367106291086654384534, −6.24358199903820219619367005977, −5.64635021183159904148624917342, −5.31548890886528691373652036289, −4.46533155092225306713593071262, −3.27860747218273468854131463134, −2.64870957228651598177273917966, −1.63787768964363617144269538442, 0, 1.63787768964363617144269538442, 2.64870957228651598177273917966, 3.27860747218273468854131463134, 4.46533155092225306713593071262, 5.31548890886528691373652036289, 5.64635021183159904148624917342, 6.24358199903820219619367005977, 7.48161600367106291086654384534, 7.88505706785551977188943828804

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