Properties

Label 2-6015-1.1-c1-0-30
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  − 2.80·2-s + 3-s + 5.85·4-s + 5-s − 2.80·6-s + 1.72·7-s − 10.8·8-s + 9-s − 2.80·10-s − 5.83·11-s + 5.85·12-s − 6.22·13-s − 4.82·14-s + 15-s + 18.5·16-s − 3.27·17-s − 2.80·18-s − 0.859·19-s + 5.85·20-s + 1.72·21-s + 16.3·22-s − 8.17·23-s − 10.8·24-s + 25-s + 17.4·26-s + 27-s + 10.0·28-s + ⋯
L(s)  = 1  − 1.98·2-s + 0.577·3-s + 2.92·4-s + 0.447·5-s − 1.14·6-s + 0.650·7-s − 3.82·8-s + 0.333·9-s − 0.886·10-s − 1.75·11-s + 1.69·12-s − 1.72·13-s − 1.28·14-s + 0.258·15-s + 4.64·16-s − 0.795·17-s − 0.660·18-s − 0.197·19-s + 1.30·20-s + 0.375·21-s + 3.48·22-s − 1.70·23-s − 2.20·24-s + 0.200·25-s + 3.42·26-s + 0.192·27-s + 1.90·28-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

\(L(1)\) \(\approx\) \(0.6377626443\)
\(L(\frac12)\) \(\approx\) \(0.6377626443\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.80T + 2T^{2} \)
7 \( 1 - 1.72T + 7T^{2} \)
11 \( 1 + 5.83T + 11T^{2} \)
13 \( 1 + 6.22T + 13T^{2} \)
17 \( 1 + 3.27T + 17T^{2} \)
19 \( 1 + 0.859T + 19T^{2} \)
23 \( 1 + 8.17T + 23T^{2} \)
29 \( 1 - 5.11T + 29T^{2} \)
31 \( 1 + 4.70T + 31T^{2} \)
37 \( 1 + 0.239T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 9.74T + 43T^{2} \)
47 \( 1 + 3.17T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 14.7T + 59T^{2} \)
61 \( 1 - 8.97T + 61T^{2} \)
67 \( 1 - 0.576T + 67T^{2} \)
71 \( 1 + 0.930T + 71T^{2} \)
73 \( 1 - 2.06T + 73T^{2} \)
79 \( 1 - 3.66T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 4.62T + 89T^{2} \)
97 \( 1 - 6.57T + 97T^{2} \)
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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

−8.107165689298970419948671073778, −7.57290656804260306017718879895, −7.25095985872548357589693387662, −6.19906691523159620773351580703, −5.47936421086598034722225899424, −4.49059277032120529814871458934, −2.96002497291085436062510524422, −2.22073547656868932527392025277, −2.07034493330074168182343813890, −0.50823845023824780065118085396, 0.50823845023824780065118085396, 2.07034493330074168182343813890, 2.22073547656868932527392025277, 2.96002497291085436062510524422, 4.49059277032120529814871458934, 5.47936421086598034722225899424, 6.19906691523159620773351580703, 7.25095985872548357589693387662, 7.57290656804260306017718879895, 8.107165689298970419948671073778

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