Properties

Label 2-6016-1.1-c1-0-10
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  − 3.18·3-s + 1.45·5-s − 4.05·7-s + 7.11·9-s + 2.02·11-s − 5.40·13-s − 4.64·15-s + 0.106·17-s − 7.30·19-s + 12.9·21-s − 1.01·23-s − 2.87·25-s − 13.0·27-s − 1.97·29-s + 1.08·31-s − 6.45·33-s − 5.92·35-s + 5.58·37-s + 17.2·39-s − 2.25·41-s − 2.62·43-s + 10.3·45-s − 47-s + 9.46·49-s − 0.339·51-s + 3.83·53-s + 2.96·55-s + ⋯
L(s)  = 1  − 1.83·3-s + 0.652·5-s − 1.53·7-s + 2.37·9-s + 0.612·11-s − 1.50·13-s − 1.19·15-s + 0.0259·17-s − 1.67·19-s + 2.81·21-s − 0.211·23-s − 0.574·25-s − 2.51·27-s − 0.366·29-s + 0.195·31-s − 1.12·33-s − 1.00·35-s + 0.918·37-s + 2.75·39-s − 0.351·41-s − 0.400·43-s + 1.54·45-s − 0.145·47-s + 1.35·49-s − 0.0476·51-s + 0.527·53-s + 0.399·55-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

\(L(1)\) \(\approx\) \(0.2735772310\)
\(L(\frac12)\) \(\approx\) \(0.2735772310\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.18T + 3T^{2} \)
5 \( 1 - 1.45T + 5T^{2} \)
7 \( 1 + 4.05T + 7T^{2} \)
11 \( 1 - 2.02T + 11T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 - 0.106T + 17T^{2} \)
19 \( 1 + 7.30T + 19T^{2} \)
23 \( 1 + 1.01T + 23T^{2} \)
29 \( 1 + 1.97T + 29T^{2} \)
31 \( 1 - 1.08T + 31T^{2} \)
37 \( 1 - 5.58T + 37T^{2} \)
41 \( 1 + 2.25T + 41T^{2} \)
43 \( 1 + 2.62T + 43T^{2} \)
53 \( 1 - 3.83T + 53T^{2} \)
59 \( 1 + 6.49T + 59T^{2} \)
61 \( 1 + 7.99T + 61T^{2} \)
67 \( 1 + 3.82T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 16.6T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−7.81491501870812196474512464359, −6.93309647646750720807620916683, −6.50728582558825875438414024236, −6.05670292439219138146666190068, −5.42443909275157040266511713801, −4.55162225741095201574933106145, −3.94416225736640173535182213274, −2.66724942501654451625375519115, −1.67318513791314866002483630839, −0.29440712644606385717967757865, 0.29440712644606385717967757865, 1.67318513791314866002483630839, 2.66724942501654451625375519115, 3.94416225736640173535182213274, 4.55162225741095201574933106145, 5.42443909275157040266511713801, 6.05670292439219138146666190068, 6.50728582558825875438414024236, 6.93309647646750720807620916683, 7.81491501870812196474512464359

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