Properties

Label 2-6015-1.1-c1-0-224
Degree $2$
Conductor $6015$
Sign $-1$
Analytic cond. $48.0300$
Root an. cond. $6.93036$
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  + 0.0493·2-s + 3-s − 1.99·4-s + 5-s + 0.0493·6-s + 2.05·7-s − 0.197·8-s + 9-s + 0.0493·10-s − 2.48·11-s − 1.99·12-s − 4.54·13-s + 0.101·14-s + 15-s + 3.98·16-s + 0.213·17-s + 0.0493·18-s + 3.24·19-s − 1.99·20-s + 2.05·21-s − 0.122·22-s − 4.81·23-s − 0.197·24-s + 25-s − 0.224·26-s + 27-s − 4.09·28-s + ⋯
L(s)  = 1  + 0.0348·2-s + 0.577·3-s − 0.998·4-s + 0.447·5-s + 0.0201·6-s + 0.774·7-s − 0.0696·8-s + 0.333·9-s + 0.0155·10-s − 0.749·11-s − 0.576·12-s − 1.26·13-s + 0.0270·14-s + 0.258·15-s + 0.996·16-s + 0.0518·17-s + 0.0116·18-s + 0.745·19-s − 0.446·20-s + 0.447·21-s − 0.0261·22-s − 1.00·23-s − 0.0402·24-s + 0.200·25-s − 0.0439·26-s + 0.192·27-s − 0.773·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6015\)    =    \(3 \cdot 5 \cdot 401\)
Sign: $-1$
Analytic conductor: \(48.0300\)
Root analytic conductor: \(6.93036\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6015,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 - T \)
401 \( 1 + T \)
good2 \( 1 - 0.0493T + 2T^{2} \)
7 \( 1 - 2.05T + 7T^{2} \)
11 \( 1 + 2.48T + 11T^{2} \)
13 \( 1 + 4.54T + 13T^{2} \)
17 \( 1 - 0.213T + 17T^{2} \)
19 \( 1 - 3.24T + 19T^{2} \)
23 \( 1 + 4.81T + 23T^{2} \)
29 \( 1 + 0.162T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 - 2.70T + 37T^{2} \)
41 \( 1 - 1.86T + 41T^{2} \)
43 \( 1 + 6.03T + 43T^{2} \)
47 \( 1 - 9.97T + 47T^{2} \)
53 \( 1 + 5.57T + 53T^{2} \)
59 \( 1 + 7.70T + 59T^{2} \)
61 \( 1 + 7.15T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 - 6.58T + 71T^{2} \)
73 \( 1 + 0.128T + 73T^{2} \)
79 \( 1 - 0.411T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 - 18.5T + 97T^{2} \)
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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

−7.76038094114860521585306628850, −7.40566640127697323913460391454, −6.16341646470352548571313829726, −5.32862746432551370009878974750, −4.87447649169374283581143451402, −4.17100386451458002858016457314, −3.18478512094667590316262315013, −2.35742340477810366551440948811, −1.41371796112523207460556335628, 0, 1.41371796112523207460556335628, 2.35742340477810366551440948811, 3.18478512094667590316262315013, 4.17100386451458002858016457314, 4.87447649169374283581143451402, 5.32862746432551370009878974750, 6.16341646470352548571313829726, 7.40566640127697323913460391454, 7.76038094114860521585306628850

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