Properties

Label 2-6015-1.1-c1-0-179
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  + 0.843·2-s − 3-s − 1.28·4-s + 5-s − 0.843·6-s − 1.98·7-s − 2.77·8-s + 9-s + 0.843·10-s + 2.08·11-s + 1.28·12-s − 0.684·13-s − 1.67·14-s − 15-s + 0.233·16-s + 4.82·17-s + 0.843·18-s − 3.22·19-s − 1.28·20-s + 1.98·21-s + 1.76·22-s − 2.25·23-s + 2.77·24-s + 25-s − 0.577·26-s − 27-s + 2.56·28-s + ⋯
L(s)  = 1  + 0.596·2-s − 0.577·3-s − 0.643·4-s + 0.447·5-s − 0.344·6-s − 0.751·7-s − 0.981·8-s + 0.333·9-s + 0.266·10-s + 0.629·11-s + 0.371·12-s − 0.189·13-s − 0.448·14-s − 0.258·15-s + 0.0583·16-s + 1.17·17-s + 0.198·18-s − 0.738·19-s − 0.287·20-s + 0.433·21-s + 0.375·22-s − 0.469·23-s + 0.566·24-s + 0.200·25-s − 0.113·26-s − 0.192·27-s + 0.483·28-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 \( 1 - 0.843T + 2T^{2} \)
7 \( 1 + 1.98T + 7T^{2} \)
11 \( 1 - 2.08T + 11T^{2} \)
13 \( 1 + 0.684T + 13T^{2} \)
17 \( 1 - 4.82T + 17T^{2} \)
19 \( 1 + 3.22T + 19T^{2} \)
23 \( 1 + 2.25T + 23T^{2} \)
29 \( 1 + 0.651T + 29T^{2} \)
31 \( 1 + 1.09T + 31T^{2} \)
37 \( 1 + 6.34T + 37T^{2} \)
41 \( 1 - 6.70T + 41T^{2} \)
43 \( 1 + 1.40T + 43T^{2} \)
47 \( 1 - 7.23T + 47T^{2} \)
53 \( 1 - 4.84T + 53T^{2} \)
59 \( 1 - 0.173T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 - 8.21T + 67T^{2} \)
71 \( 1 - 5.11T + 71T^{2} \)
73 \( 1 + 2.02T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 - 4.15T + 83T^{2} \)
89 \( 1 + 4.45T + 89T^{2} \)
97 \( 1 + 6.46T + 97T^{2} \)
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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

−7.63457406513707215060618956708, −6.71136048021505207851396910959, −6.14806366060328768220077151296, −5.58392665879699939479831342365, −4.92790536818000736602955940378, −4.01602603656850504715033382613, −3.51352175622727999954641471154, −2.47915645288322922559077987709, −1.18895313516457969368769796168, 0, 1.18895313516457969368769796168, 2.47915645288322922559077987709, 3.51352175622727999954641471154, 4.01602603656850504715033382613, 4.92790536818000736602955940378, 5.58392665879699939479831342365, 6.14806366060328768220077151296, 6.71136048021505207851396910959, 7.63457406513707215060618956708

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