Properties

Label 2-6015-1.1-c1-0-254
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  + 1.03·2-s + 3-s − 0.924·4-s + 5-s + 1.03·6-s + 2.93·7-s − 3.03·8-s + 9-s + 1.03·10-s + 1.72·11-s − 0.924·12-s − 3.10·13-s + 3.04·14-s + 15-s − 1.29·16-s − 2.83·17-s + 1.03·18-s − 2.66·19-s − 0.924·20-s + 2.93·21-s + 1.78·22-s − 9.34·23-s − 3.03·24-s + 25-s − 3.22·26-s + 27-s − 2.71·28-s + ⋯
L(s)  = 1  + 0.733·2-s + 0.577·3-s − 0.462·4-s + 0.447·5-s + 0.423·6-s + 1.11·7-s − 1.07·8-s + 0.333·9-s + 0.327·10-s + 0.519·11-s − 0.266·12-s − 0.862·13-s + 0.814·14-s + 0.258·15-s − 0.324·16-s − 0.686·17-s + 0.244·18-s − 0.610·19-s − 0.206·20-s + 0.641·21-s + 0.380·22-s − 1.94·23-s − 0.619·24-s + 0.200·25-s − 0.632·26-s + 0.192·27-s − 0.513·28-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 \( 1 - 1.03T + 2T^{2} \)
7 \( 1 - 2.93T + 7T^{2} \)
11 \( 1 - 1.72T + 11T^{2} \)
13 \( 1 + 3.10T + 13T^{2} \)
17 \( 1 + 2.83T + 17T^{2} \)
19 \( 1 + 2.66T + 19T^{2} \)
23 \( 1 + 9.34T + 23T^{2} \)
29 \( 1 + 9.24T + 29T^{2} \)
31 \( 1 + 1.99T + 31T^{2} \)
37 \( 1 + 6.47T + 37T^{2} \)
41 \( 1 + 0.0256T + 41T^{2} \)
43 \( 1 + 7.79T + 43T^{2} \)
47 \( 1 + 0.767T + 47T^{2} \)
53 \( 1 - 3.39T + 53T^{2} \)
59 \( 1 - 8.73T + 59T^{2} \)
61 \( 1 + 0.796T + 61T^{2} \)
67 \( 1 - 5.13T + 67T^{2} \)
71 \( 1 - 3.65T + 71T^{2} \)
73 \( 1 + 9.44T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + 0.994T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−7.75217553880875556631535459881, −7.02910573052086255302534878224, −6.10513150538070335344051495961, −5.44299341226535116746446734517, −4.71353542476793580678219303978, −4.12999168889653509770183163927, −3.45776150160508342479291396144, −2.22489535849249656783148959117, −1.77200369790137327358489416019, 0, 1.77200369790137327358489416019, 2.22489535849249656783148959117, 3.45776150160508342479291396144, 4.12999168889653509770183163927, 4.71353542476793580678219303978, 5.44299341226535116746446734517, 6.10513150538070335344051495961, 7.02910573052086255302534878224, 7.75217553880875556631535459881

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