Properties

Label 2-4015-1.1-c1-0-220
Degree $2$
Conductor $4015$
Sign $-1$
Analytic cond. $32.0599$
Root an. cond. $5.66214$
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  + 2.28·2-s − 0.471·3-s + 3.23·4-s − 5-s − 1.07·6-s + 0.586·7-s + 2.81·8-s − 2.77·9-s − 2.28·10-s + 11-s − 1.52·12-s − 5.54·13-s + 1.34·14-s + 0.471·15-s − 0.0236·16-s + 3.19·17-s − 6.35·18-s + 3.32·19-s − 3.23·20-s − 0.276·21-s + 2.28·22-s − 7.54·23-s − 1.32·24-s + 25-s − 12.6·26-s + 2.72·27-s + 1.89·28-s + ⋯
L(s)  = 1  + 1.61·2-s − 0.272·3-s + 1.61·4-s − 0.447·5-s − 0.440·6-s + 0.221·7-s + 0.995·8-s − 0.925·9-s − 0.723·10-s + 0.301·11-s − 0.439·12-s − 1.53·13-s + 0.358·14-s + 0.121·15-s − 0.00590·16-s + 0.773·17-s − 1.49·18-s + 0.761·19-s − 0.722·20-s − 0.0604·21-s + 0.487·22-s − 1.57·23-s − 0.271·24-s + 0.200·25-s − 2.48·26-s + 0.524·27-s + 0.358·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4015\)    =    \(5 \cdot 11 \cdot 73\)
Sign: $-1$
Analytic conductor: \(32.0599\)
Root analytic conductor: \(5.66214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4015,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 + T \)
11 \( 1 - T \)
73 \( 1 - T \)
good2 \( 1 - 2.28T + 2T^{2} \)
3 \( 1 + 0.471T + 3T^{2} \)
7 \( 1 - 0.586T + 7T^{2} \)
13 \( 1 + 5.54T + 13T^{2} \)
17 \( 1 - 3.19T + 17T^{2} \)
19 \( 1 - 3.32T + 19T^{2} \)
23 \( 1 + 7.54T + 23T^{2} \)
29 \( 1 - 10.7T + 29T^{2} \)
31 \( 1 - 0.332T + 31T^{2} \)
37 \( 1 + 4.54T + 37T^{2} \)
41 \( 1 + 8.04T + 41T^{2} \)
43 \( 1 + 8.68T + 43T^{2} \)
47 \( 1 + 4.96T + 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 - 9.53T + 59T^{2} \)
61 \( 1 + 6.99T + 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 - 8.63T + 71T^{2} \)
79 \( 1 + 9.24T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 7.44T + 89T^{2} \)
97 \( 1 - 6.57T + 97T^{2} \)
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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

−7.945266219638310123292314732263, −7.06071229296444259353594255830, −6.39990207521645461637767124975, −5.65306462951231642849198069246, −4.90774608088815663798220938899, −4.56464569176210303329178990813, −3.33055134608603305103194242798, −2.98643882140629983289184706897, −1.81012246758245517864048738028, 0, 1.81012246758245517864048738028, 2.98643882140629983289184706897, 3.33055134608603305103194242798, 4.56464569176210303329178990813, 4.90774608088815663798220938899, 5.65306462951231642849198069246, 6.39990207521645461637767124975, 7.06071229296444259353594255830, 7.945266219638310123292314732263

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