Properties

Label 2-6015-1.1-c1-0-165
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.09·2-s + 3-s + 2.39·4-s + 5-s + 2.09·6-s − 1.39·7-s + 0.834·8-s + 9-s + 2.09·10-s + 3.33·11-s + 2.39·12-s + 5.70·13-s − 2.92·14-s + 15-s − 3.04·16-s − 2.75·17-s + 2.09·18-s − 3.13·19-s + 2.39·20-s − 1.39·21-s + 6.99·22-s + 6.71·23-s + 0.834·24-s + 25-s + 11.9·26-s + 27-s − 3.34·28-s + ⋯
L(s)  = 1  + 1.48·2-s + 0.577·3-s + 1.19·4-s + 0.447·5-s + 0.856·6-s − 0.527·7-s + 0.294·8-s + 0.333·9-s + 0.663·10-s + 1.00·11-s + 0.692·12-s + 1.58·13-s − 0.782·14-s + 0.258·15-s − 0.761·16-s − 0.667·17-s + 0.494·18-s − 0.719·19-s + 0.536·20-s − 0.304·21-s + 1.49·22-s + 1.40·23-s + 0.170·24-s + 0.200·25-s + 2.34·26-s + 0.192·27-s − 0.632·28-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: \(0\)
Selberg data: \((2,\ 6015,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.693343136\)
\(L(\frac12)\) \(\approx\) \(6.693343136\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.09T + 2T^{2} \)
7 \( 1 + 1.39T + 7T^{2} \)
11 \( 1 - 3.33T + 11T^{2} \)
13 \( 1 - 5.70T + 13T^{2} \)
17 \( 1 + 2.75T + 17T^{2} \)
19 \( 1 + 3.13T + 19T^{2} \)
23 \( 1 - 6.71T + 23T^{2} \)
29 \( 1 - 3.04T + 29T^{2} \)
31 \( 1 - 9.55T + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 - 3.99T + 41T^{2} \)
43 \( 1 - 1.58T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 5.35T + 53T^{2} \)
59 \( 1 + 3.37T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 - 3.88T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 2.46T + 79T^{2} \)
83 \( 1 - 5.54T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−8.173132123483470805815232739461, −6.80291988819102305121346631280, −6.52087383313922542937764570538, −6.09152736019509918712429837827, −4.99862748953386544207075419951, −4.39894391018011587055719721091, −3.61243446379562210891931308743, −3.11867245233017301530324432104, −2.21603337691121290145273288902, −1.13995377133954938655384621026, 1.13995377133954938655384621026, 2.21603337691121290145273288902, 3.11867245233017301530324432104, 3.61243446379562210891931308743, 4.39894391018011587055719721091, 4.99862748953386544207075419951, 6.09152736019509918712429837827, 6.52087383313922542937764570538, 6.80291988819102305121346631280, 8.173132123483470805815232739461

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