Properties

Label 2-4015-1.1-c1-0-64
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  − 2.31·2-s − 1.68·3-s + 3.34·4-s − 5-s + 3.89·6-s − 2.49·7-s − 3.10·8-s − 0.162·9-s + 2.31·10-s − 11-s − 5.63·12-s − 5.27·13-s + 5.76·14-s + 1.68·15-s + 0.495·16-s + 2.07·17-s + 0.376·18-s − 2.21·19-s − 3.34·20-s + 4.20·21-s + 2.31·22-s − 3.45·23-s + 5.23·24-s + 25-s + 12.2·26-s + 5.32·27-s − 8.34·28-s + ⋯
L(s)  = 1  − 1.63·2-s − 0.972·3-s + 1.67·4-s − 0.447·5-s + 1.58·6-s − 0.942·7-s − 1.09·8-s − 0.0543·9-s + 0.731·10-s − 0.301·11-s − 1.62·12-s − 1.46·13-s + 1.54·14-s + 0.434·15-s + 0.123·16-s + 0.504·17-s + 0.0887·18-s − 0.509·19-s − 0.747·20-s + 0.916·21-s + 0.492·22-s − 0.720·23-s + 1.06·24-s + 0.200·25-s + 2.39·26-s + 1.02·27-s − 1.57·28-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 \( 1 + 2.31T + 2T^{2} \)
3 \( 1 + 1.68T + 3T^{2} \)
7 \( 1 + 2.49T + 7T^{2} \)
13 \( 1 + 5.27T + 13T^{2} \)
17 \( 1 - 2.07T + 17T^{2} \)
19 \( 1 + 2.21T + 19T^{2} \)
23 \( 1 + 3.45T + 23T^{2} \)
29 \( 1 - 1.15T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 7.87T + 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 2.08T + 43T^{2} \)
47 \( 1 + 7.81T + 47T^{2} \)
53 \( 1 - 8.46T + 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + 3.66T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 2.46T + 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
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

−8.230213682650533060583299340312, −7.37111621097015194248933826775, −6.81423135562917588872883672419, −6.17893344038729787162618878971, −5.26915069122669909775753416122, −4.38078733953032656415667258641, −3.04946035051038380910185238011, −2.25086148532179030284347300181, −0.75267552993475557122500231472, 0, 0.75267552993475557122500231472, 2.25086148532179030284347300181, 3.04946035051038380910185238011, 4.38078733953032656415667258641, 5.26915069122669909775753416122, 6.17893344038729787162618878971, 6.81423135562917588872883672419, 7.37111621097015194248933826775, 8.230213682650533060583299340312

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