Properties

Label 2-6015-1.1-c1-0-120
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  − 1.45·2-s + 3-s + 0.129·4-s − 5-s − 1.45·6-s + 4.22·7-s + 2.72·8-s + 9-s + 1.45·10-s + 5.03·11-s + 0.129·12-s + 2.78·13-s − 6.16·14-s − 15-s − 4.24·16-s − 2.25·17-s − 1.45·18-s − 4.42·19-s − 0.129·20-s + 4.22·21-s − 7.35·22-s + 3.01·23-s + 2.72·24-s + 25-s − 4.06·26-s + 27-s + 0.548·28-s + ⋯
L(s)  = 1  − 1.03·2-s + 0.577·3-s + 0.0648·4-s − 0.447·5-s − 0.595·6-s + 1.59·7-s + 0.964·8-s + 0.333·9-s + 0.461·10-s + 1.51·11-s + 0.0374·12-s + 0.772·13-s − 1.64·14-s − 0.258·15-s − 1.06·16-s − 0.547·17-s − 0.343·18-s − 1.01·19-s − 0.0290·20-s + 0.922·21-s − 1.56·22-s + 0.628·23-s + 0.557·24-s + 0.200·25-s − 0.797·26-s + 0.192·27-s + 0.103·28-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

\(L(1)\) \(\approx\) \(1.881269677\)
\(L(\frac12)\) \(\approx\) \(1.881269677\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.45T + 2T^{2} \)
7 \( 1 - 4.22T + 7T^{2} \)
11 \( 1 - 5.03T + 11T^{2} \)
13 \( 1 - 2.78T + 13T^{2} \)
17 \( 1 + 2.25T + 17T^{2} \)
19 \( 1 + 4.42T + 19T^{2} \)
23 \( 1 - 3.01T + 23T^{2} \)
29 \( 1 - 4.42T + 29T^{2} \)
31 \( 1 - 7.05T + 31T^{2} \)
37 \( 1 - 4.87T + 37T^{2} \)
41 \( 1 + 2.84T + 41T^{2} \)
43 \( 1 - 9.33T + 43T^{2} \)
47 \( 1 + 3.28T + 47T^{2} \)
53 \( 1 + 5.95T + 53T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 + 7.34T + 61T^{2} \)
67 \( 1 + 2.58T + 67T^{2} \)
71 \( 1 + 3.18T + 71T^{2} \)
73 \( 1 - 8.59T + 73T^{2} \)
79 \( 1 + 6.03T + 79T^{2} \)
83 \( 1 + 0.267T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 8.87T + 97T^{2} \)
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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

−8.307968117632935658858323192438, −7.71841060861107787796863203007, −6.89580741819445325798807519634, −6.25221477047132127571923292677, −4.88599969336362050336381534703, −4.36500161935296384215144691715, −3.85921637248865735328961770275, −2.48123187313833568385887602148, −1.50010993724036640762678212388, −0.955570314961024798020639222751, 0.955570314961024798020639222751, 1.50010993724036640762678212388, 2.48123187313833568385887602148, 3.85921637248865735328961770275, 4.36500161935296384215144691715, 4.88599969336362050336381534703, 6.25221477047132127571923292677, 6.89580741819445325798807519634, 7.71841060861107787796863203007, 8.307968117632935658858323192438

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