Properties

Label 2-6016-1.1-c1-0-37
Degree $2$
Conductor $6016$
Sign $1$
Analytic cond. $48.0380$
Root an. cond. $6.93094$
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  − 0.319·3-s + 2.22·5-s + 0.484·7-s − 2.89·9-s − 3.07·11-s − 5.65·13-s − 0.710·15-s + 1.35·17-s − 2.21·19-s − 0.154·21-s + 1.53·23-s − 0.0565·25-s + 1.88·27-s − 3.86·29-s + 5.33·31-s + 0.982·33-s + 1.07·35-s + 6.61·37-s + 1.80·39-s − 0.811·41-s + 0.404·43-s − 6.44·45-s − 47-s − 6.76·49-s − 0.432·51-s + 9.23·53-s − 6.83·55-s + ⋯
L(s)  = 1  − 0.184·3-s + 0.994·5-s + 0.183·7-s − 0.965·9-s − 0.926·11-s − 1.56·13-s − 0.183·15-s + 0.328·17-s − 0.509·19-s − 0.0337·21-s + 0.319·23-s − 0.0113·25-s + 0.362·27-s − 0.717·29-s + 0.957·31-s + 0.170·33-s + 0.182·35-s + 1.08·37-s + 0.289·39-s − 0.126·41-s + 0.0616·43-s − 0.960·45-s − 0.145·47-s − 0.966·49-s − 0.0606·51-s + 1.26·53-s − 0.921·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6016\)    =    \(2^{7} \cdot 47\)
Sign: $1$
Analytic conductor: \(48.0380\)
Root analytic conductor: \(6.93094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.522759586\)
\(L(\frac12)\) \(\approx\) \(1.522759586\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
47 \( 1 + T \)
good3 \( 1 + 0.319T + 3T^{2} \)
5 \( 1 - 2.22T + 5T^{2} \)
7 \( 1 - 0.484T + 7T^{2} \)
11 \( 1 + 3.07T + 11T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 - 1.35T + 17T^{2} \)
19 \( 1 + 2.21T + 19T^{2} \)
23 \( 1 - 1.53T + 23T^{2} \)
29 \( 1 + 3.86T + 29T^{2} \)
31 \( 1 - 5.33T + 31T^{2} \)
37 \( 1 - 6.61T + 37T^{2} \)
41 \( 1 + 0.811T + 41T^{2} \)
43 \( 1 - 0.404T + 43T^{2} \)
53 \( 1 - 9.23T + 53T^{2} \)
59 \( 1 - 1.47T + 59T^{2} \)
61 \( 1 - 2.46T + 61T^{2} \)
67 \( 1 - 13.7T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 3.68T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 5.17T + 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−8.019023906803399317311743764011, −7.46716486058115196539371016301, −6.50545239612849589813590466560, −5.90980233626355027303446529827, −5.13993416463790579313513128263, −4.84245051863510595905867346126, −3.51598217855072992614994799372, −2.43691234487269414100794745580, −2.22558357669153795086935429850, −0.61884588254630013703553648753, 0.61884588254630013703553648753, 2.22558357669153795086935429850, 2.43691234487269414100794745580, 3.51598217855072992614994799372, 4.84245051863510595905867346126, 5.13993416463790579313513128263, 5.90980233626355027303446529827, 6.50545239612849589813590466560, 7.46716486058115196539371016301, 8.019023906803399317311743764011

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