Properties

Label 2-6015-1.1-c1-0-29
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  + 0.288·2-s − 3-s − 1.91·4-s + 5-s − 0.288·6-s − 0.447·7-s − 1.13·8-s + 9-s + 0.288·10-s − 3.45·11-s + 1.91·12-s + 5.09·13-s − 0.129·14-s − 15-s + 3.50·16-s − 3.00·17-s + 0.288·18-s − 6.49·19-s − 1.91·20-s + 0.447·21-s − 0.997·22-s − 0.708·23-s + 1.13·24-s + 25-s + 1.47·26-s − 27-s + 0.858·28-s + ⋯
L(s)  = 1  + 0.204·2-s − 0.577·3-s − 0.958·4-s + 0.447·5-s − 0.117·6-s − 0.169·7-s − 0.399·8-s + 0.333·9-s + 0.0913·10-s − 1.04·11-s + 0.553·12-s + 1.41·13-s − 0.0345·14-s − 0.258·15-s + 0.876·16-s − 0.727·17-s + 0.0680·18-s − 1.49·19-s − 0.428·20-s + 0.0977·21-s − 0.212·22-s − 0.147·23-s + 0.230·24-s + 0.200·25-s + 0.288·26-s − 0.192·27-s + 0.162·28-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

\(L(1)\) \(\approx\) \(0.9410509432\)
\(L(\frac12)\) \(\approx\) \(0.9410509432\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.288T + 2T^{2} \)
7 \( 1 + 0.447T + 7T^{2} \)
11 \( 1 + 3.45T + 11T^{2} \)
13 \( 1 - 5.09T + 13T^{2} \)
17 \( 1 + 3.00T + 17T^{2} \)
19 \( 1 + 6.49T + 19T^{2} \)
23 \( 1 + 0.708T + 23T^{2} \)
29 \( 1 + 6.50T + 29T^{2} \)
31 \( 1 + 0.921T + 31T^{2} \)
37 \( 1 - 3.37T + 37T^{2} \)
41 \( 1 - 4.39T + 41T^{2} \)
43 \( 1 - 2.87T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 5.23T + 53T^{2} \)
59 \( 1 - 1.40T + 59T^{2} \)
61 \( 1 + 9.14T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 - 1.07T + 71T^{2} \)
73 \( 1 - 3.14T + 73T^{2} \)
79 \( 1 + 5.49T + 79T^{2} \)
83 \( 1 - 3.57T + 83T^{2} \)
89 \( 1 - 0.624T + 89T^{2} \)
97 \( 1 - 0.895T + 97T^{2} \)
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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

−8.178001558630848802502983878358, −7.38381850198596419680040392009, −6.32194190710019673037785551698, −5.91789322640259438919847260449, −5.31075212202259472466059862637, −4.36928371809585158612796127378, −3.95691513769609894443482247852, −2.83552621629943890314241823752, −1.77973099028443188417987758733, −0.50781199503161542161999514209, 0.50781199503161542161999514209, 1.77973099028443188417987758733, 2.83552621629943890314241823752, 3.95691513769609894443482247852, 4.36928371809585158612796127378, 5.31075212202259472466059862637, 5.91789322640259438919847260449, 6.32194190710019673037785551698, 7.38381850198596419680040392009, 8.178001558630848802502983878358

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