Properties

Label 2-6016-1.1-c1-0-93
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  + 2.61·3-s − 1.63·5-s + 3.46·7-s + 3.85·9-s + 1.65·11-s − 2.82·13-s − 4.27·15-s + 6.37·17-s + 3.98·19-s + 9.07·21-s − 2.04·23-s − 2.33·25-s + 2.25·27-s − 0.0143·29-s + 7.31·31-s + 4.32·33-s − 5.65·35-s − 4.31·37-s − 7.38·39-s + 10.1·41-s − 1.72·43-s − 6.29·45-s − 47-s + 5.00·49-s + 16.6·51-s + 5.63·53-s − 2.69·55-s + ⋯
L(s)  = 1  + 1.51·3-s − 0.729·5-s + 1.30·7-s + 1.28·9-s + 0.498·11-s − 0.782·13-s − 1.10·15-s + 1.54·17-s + 0.915·19-s + 1.98·21-s − 0.426·23-s − 0.467·25-s + 0.433·27-s − 0.00265·29-s + 1.31·31-s + 0.753·33-s − 0.955·35-s − 0.709·37-s − 1.18·39-s + 1.58·41-s − 0.263·43-s − 0.938·45-s − 0.145·47-s + 0.715·49-s + 2.33·51-s + 0.774·53-s − 0.363·55-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

\(L(1)\) \(\approx\) \(3.996124197\)
\(L(\frac12)\) \(\approx\) \(3.996124197\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 3T^{2} \)
5 \( 1 + 1.63T + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 1.65T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 - 6.37T + 17T^{2} \)
19 \( 1 - 3.98T + 19T^{2} \)
23 \( 1 + 2.04T + 23T^{2} \)
29 \( 1 + 0.0143T + 29T^{2} \)
31 \( 1 - 7.31T + 31T^{2} \)
37 \( 1 + 4.31T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 1.72T + 43T^{2} \)
53 \( 1 - 5.63T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 1.84T + 61T^{2} \)
67 \( 1 - 3.19T + 67T^{2} \)
71 \( 1 + 1.78T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 - 7.77T + 83T^{2} \)
89 \( 1 + 0.300T + 89T^{2} \)
97 \( 1 - 3.62T + 97T^{2} \)
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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

−7.997941641356191025845373255754, −7.68435554437467981265582664939, −7.12811705742385968389568786127, −5.86811520899257565668191422995, −4.99052992526068317701730435807, −4.27206794506020826555738708359, −3.57569974946720765291269438427, −2.86002246953585681065948619165, −1.94674055022828562120700970610, −1.05158084530840030377964285587, 1.05158084530840030377964285587, 1.94674055022828562120700970610, 2.86002246953585681065948619165, 3.57569974946720765291269438427, 4.27206794506020826555738708359, 4.99052992526068317701730435807, 5.86811520899257565668191422995, 7.12811705742385968389568786127, 7.68435554437467981265582664939, 7.997941641356191025845373255754

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