Properties

Label 2-6016-1.1-c1-0-59
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  − 2.08·3-s − 4.13·5-s + 1.67·7-s + 1.33·9-s + 2.59·11-s + 6.34·13-s + 8.61·15-s + 6.39·17-s − 0.134·19-s − 3.49·21-s + 8.53·23-s + 12.0·25-s + 3.46·27-s + 0.201·29-s − 0.349·31-s − 5.39·33-s − 6.93·35-s + 5.26·37-s − 13.2·39-s + 5.79·41-s − 8.54·43-s − 5.53·45-s − 47-s − 4.18·49-s − 13.3·51-s − 8.38·53-s − 10.7·55-s + ⋯
L(s)  = 1  − 1.20·3-s − 1.84·5-s + 0.634·7-s + 0.446·9-s + 0.781·11-s + 1.75·13-s + 2.22·15-s + 1.55·17-s − 0.0307·19-s − 0.762·21-s + 1.77·23-s + 2.41·25-s + 0.665·27-s + 0.0374·29-s − 0.0627·31-s − 0.939·33-s − 1.17·35-s + 0.865·37-s − 2.11·39-s + 0.904·41-s − 1.30·43-s − 0.825·45-s − 0.145·47-s − 0.597·49-s − 1.86·51-s − 1.15·53-s − 1.44·55-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

\(L(1)\) \(\approx\) \(1.242588275\)
\(L(\frac12)\) \(\approx\) \(1.242588275\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.08T + 3T^{2} \)
5 \( 1 + 4.13T + 5T^{2} \)
7 \( 1 - 1.67T + 7T^{2} \)
11 \( 1 - 2.59T + 11T^{2} \)
13 \( 1 - 6.34T + 13T^{2} \)
17 \( 1 - 6.39T + 17T^{2} \)
19 \( 1 + 0.134T + 19T^{2} \)
23 \( 1 - 8.53T + 23T^{2} \)
29 \( 1 - 0.201T + 29T^{2} \)
31 \( 1 + 0.349T + 31T^{2} \)
37 \( 1 - 5.26T + 37T^{2} \)
41 \( 1 - 5.79T + 41T^{2} \)
43 \( 1 + 8.54T + 43T^{2} \)
53 \( 1 + 8.38T + 53T^{2} \)
59 \( 1 - 8.24T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 - 2.18T + 67T^{2} \)
71 \( 1 - 4.61T + 71T^{2} \)
73 \( 1 - 7.54T + 73T^{2} \)
79 \( 1 - 0.107T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + 2.22T + 89T^{2} \)
97 \( 1 - 6.89T + 97T^{2} \)
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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

−8.041776249367484061730467092274, −7.39762884392144960056227267805, −6.59290628551123147579719600138, −6.02658400945823616294587489600, −5.04863975048101051614299605623, −4.59906197549424084537090654618, −3.60500945026133715835645582873, −3.25890691236989680528726187220, −1.23583050240220888187525298306, −0.77345429979266238389041488810, 0.77345429979266238389041488810, 1.23583050240220888187525298306, 3.25890691236989680528726187220, 3.60500945026133715835645582873, 4.59906197549424084537090654618, 5.04863975048101051614299605623, 6.02658400945823616294587489600, 6.59290628551123147579719600138, 7.39762884392144960056227267805, 8.041776249367484061730467092274

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