Properties

Label 2-6015-1.1-c1-0-18
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  − 0.762·2-s + 3-s − 1.41·4-s + 5-s − 0.762·6-s − 2.97·7-s + 2.60·8-s + 9-s − 0.762·10-s − 4.64·11-s − 1.41·12-s − 0.555·13-s + 2.27·14-s + 15-s + 0.846·16-s − 0.288·17-s − 0.762·18-s − 5.60·19-s − 1.41·20-s − 2.97·21-s + 3.54·22-s − 5.10·23-s + 2.60·24-s + 25-s + 0.423·26-s + 27-s + 4.22·28-s + ⋯
L(s)  = 1  − 0.539·2-s + 0.577·3-s − 0.709·4-s + 0.447·5-s − 0.311·6-s − 1.12·7-s + 0.921·8-s + 0.333·9-s − 0.241·10-s − 1.40·11-s − 0.409·12-s − 0.154·13-s + 0.607·14-s + 0.258·15-s + 0.211·16-s − 0.0700·17-s − 0.179·18-s − 1.28·19-s − 0.317·20-s − 0.650·21-s + 0.755·22-s − 1.06·23-s + 0.532·24-s + 0.200·25-s + 0.0830·26-s + 0.192·27-s + 0.798·28-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

\(L(1)\) \(\approx\) \(0.6911371749\)
\(L(\frac12)\) \(\approx\) \(0.6911371749\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.762T + 2T^{2} \)
7 \( 1 + 2.97T + 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + 0.555T + 13T^{2} \)
17 \( 1 + 0.288T + 17T^{2} \)
19 \( 1 + 5.60T + 19T^{2} \)
23 \( 1 + 5.10T + 23T^{2} \)
29 \( 1 + 1.23T + 29T^{2} \)
31 \( 1 + 8.60T + 31T^{2} \)
37 \( 1 - 4.18T + 37T^{2} \)
41 \( 1 + 2.39T + 41T^{2} \)
43 \( 1 + 3.62T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 - 5.49T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 - 7.12T + 67T^{2} \)
71 \( 1 + 6.30T + 71T^{2} \)
73 \( 1 + 9.03T + 73T^{2} \)
79 \( 1 - 6.75T + 79T^{2} \)
83 \( 1 + 0.908T + 83T^{2} \)
89 \( 1 + 7.61T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−8.322583525181762907956876041884, −7.47000156976811273088663489673, −6.89650913087837029634462483917, −5.84308008459940570382668625525, −5.33665939115725471096047421183, −4.26186548575226985422371001379, −3.69054922640615452691896351373, −2.61357826917839429192619341116, −1.97286842448247460646214182130, −0.43922693826925267217165737435, 0.43922693826925267217165737435, 1.97286842448247460646214182130, 2.61357826917839429192619341116, 3.69054922640615452691896351373, 4.26186548575226985422371001379, 5.33665939115725471096047421183, 5.84308008459940570382668625525, 6.89650913087837029634462483917, 7.47000156976811273088663489673, 8.322583525181762907956876041884

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