Properties

Label 2-4016-1.1-c1-0-101
Degree $2$
Conductor $4016$
Sign $-1$
Analytic cond. $32.0679$
Root an. cond. $5.66285$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.20·3-s − 1.79·5-s + 2.78·7-s − 1.54·9-s − 0.183·11-s − 4.03·13-s − 2.17·15-s + 1.33·17-s + 0.976·19-s + 3.36·21-s + 6.76·23-s − 1.76·25-s − 5.48·27-s − 3.70·29-s − 5.56·31-s − 0.221·33-s − 5.01·35-s + 2.65·37-s − 4.87·39-s − 10.1·41-s + 2.03·43-s + 2.78·45-s − 11.3·47-s + 0.768·49-s + 1.60·51-s + 12.2·53-s + 0.330·55-s + ⋯
L(s)  = 1  + 0.696·3-s − 0.804·5-s + 1.05·7-s − 0.515·9-s − 0.0553·11-s − 1.12·13-s − 0.560·15-s + 0.322·17-s + 0.224·19-s + 0.733·21-s + 1.40·23-s − 0.352·25-s − 1.05·27-s − 0.687·29-s − 0.998·31-s − 0.0385·33-s − 0.847·35-s + 0.436·37-s − 0.780·39-s − 1.58·41-s + 0.309·43-s + 0.414·45-s − 1.65·47-s + 0.109·49-s + 0.224·51-s + 1.68·53-s + 0.0445·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4016\)    =    \(2^{4} \cdot 251\)
Sign: $-1$
Analytic conductor: \(32.0679\)
Root analytic conductor: \(5.66285\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4016,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 + T \)
good3 \( 1 - 1.20T + 3T^{2} \)
5 \( 1 + 1.79T + 5T^{2} \)
7 \( 1 - 2.78T + 7T^{2} \)
11 \( 1 + 0.183T + 11T^{2} \)
13 \( 1 + 4.03T + 13T^{2} \)
17 \( 1 - 1.33T + 17T^{2} \)
19 \( 1 - 0.976T + 19T^{2} \)
23 \( 1 - 6.76T + 23T^{2} \)
29 \( 1 + 3.70T + 29T^{2} \)
31 \( 1 + 5.56T + 31T^{2} \)
37 \( 1 - 2.65T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 2.03T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 2.65T + 61T^{2} \)
67 \( 1 + 1.94T + 67T^{2} \)
71 \( 1 + 5.71T + 71T^{2} \)
73 \( 1 + 8.31T + 73T^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + 8.57T + 83T^{2} \)
89 \( 1 - 0.496T + 89T^{2} \)
97 \( 1 + 0.552T + 97T^{2} \)
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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

−8.146596073907495017903494662463, −7.43133219221860934402988519536, −7.01496275981868554209835656012, −5.57841781776413019463792029615, −5.10015815685618256470235323303, −4.19242243579576732770840175382, −3.36270147215778036426060246630, −2.56410964652605455620336454075, −1.55952005505326650046201567769, 0, 1.55952005505326650046201567769, 2.56410964652605455620336454075, 3.36270147215778036426060246630, 4.19242243579576732770840175382, 5.10015815685618256470235323303, 5.57841781776413019463792029615, 7.01496275981868554209835656012, 7.43133219221860934402988519536, 8.146596073907495017903494662463

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