Properties

Label 2-6015-1.1-c1-0-219
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  + 2.23·2-s + 3-s + 2.97·4-s + 5-s + 2.23·6-s + 2.98·7-s + 2.18·8-s + 9-s + 2.23·10-s + 2.98·11-s + 2.97·12-s − 1.82·13-s + 6.65·14-s + 15-s − 1.08·16-s + 1.87·17-s + 2.23·18-s + 6.06·19-s + 2.97·20-s + 2.98·21-s + 6.66·22-s + 4.90·23-s + 2.18·24-s + 25-s − 4.06·26-s + 27-s + 8.89·28-s + ⋯
L(s)  = 1  + 1.57·2-s + 0.577·3-s + 1.48·4-s + 0.447·5-s + 0.910·6-s + 1.12·7-s + 0.772·8-s + 0.333·9-s + 0.705·10-s + 0.901·11-s + 0.859·12-s − 0.505·13-s + 1.77·14-s + 0.258·15-s − 0.271·16-s + 0.455·17-s + 0.525·18-s + 1.39·19-s + 0.666·20-s + 0.651·21-s + 1.42·22-s + 1.02·23-s + 0.445·24-s + 0.200·25-s − 0.797·26-s + 0.192·27-s + 1.68·28-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

\(L(1)\) \(\approx\) \(8.387760050\)
\(L(\frac12)\) \(\approx\) \(8.387760050\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T + 2T^{2} \)
7 \( 1 - 2.98T + 7T^{2} \)
11 \( 1 - 2.98T + 11T^{2} \)
13 \( 1 + 1.82T + 13T^{2} \)
17 \( 1 - 1.87T + 17T^{2} \)
19 \( 1 - 6.06T + 19T^{2} \)
23 \( 1 - 4.90T + 23T^{2} \)
29 \( 1 + 6.71T + 29T^{2} \)
31 \( 1 + 3.30T + 31T^{2} \)
37 \( 1 - 6.69T + 37T^{2} \)
41 \( 1 + 6.95T + 41T^{2} \)
43 \( 1 + 8.22T + 43T^{2} \)
47 \( 1 + 9.34T + 47T^{2} \)
53 \( 1 + 7.74T + 53T^{2} \)
59 \( 1 - 7.61T + 59T^{2} \)
61 \( 1 - 8.45T + 61T^{2} \)
67 \( 1 - 3.40T + 67T^{2} \)
71 \( 1 + 9.29T + 71T^{2} \)
73 \( 1 + 6.53T + 73T^{2} \)
79 \( 1 - 0.979T + 79T^{2} \)
83 \( 1 - 3.36T + 83T^{2} \)
89 \( 1 + 0.0390T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−7.86015423631544541695992049168, −7.20330695307433338489030561577, −6.58695793217842952331448100102, −5.61917344626334224406314836634, −5.09355100608252335158268220075, −4.57444731964191039051410830753, −3.58462458700668696813991076758, −3.12098982497916079915243234463, −2.04656615266516192360681310485, −1.36897517215645334159473803549, 1.36897517215645334159473803549, 2.04656615266516192360681310485, 3.12098982497916079915243234463, 3.58462458700668696813991076758, 4.57444731964191039051410830753, 5.09355100608252335158268220075, 5.61917344626334224406314836634, 6.58695793217842952331448100102, 7.20330695307433338489030561577, 7.86015423631544541695992049168

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