Properties

Label 2-4016-1.1-c1-0-82
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  − 3.07·3-s + 2.30·5-s + 0.712·7-s + 6.48·9-s − 1.51·11-s − 0.395·13-s − 7.08·15-s − 0.429·17-s + 3.54·19-s − 2.19·21-s − 2.70·23-s + 0.299·25-s − 10.7·27-s + 0.745·29-s − 6.64·31-s + 4.65·33-s + 1.63·35-s + 3.72·37-s + 1.21·39-s − 9.89·41-s − 2.82·43-s + 14.9·45-s + 3.59·47-s − 6.49·49-s + 1.32·51-s − 1.92·53-s − 3.47·55-s + ⋯
L(s)  = 1  − 1.77·3-s + 1.02·5-s + 0.269·7-s + 2.16·9-s − 0.455·11-s − 0.109·13-s − 1.83·15-s − 0.104·17-s + 0.814·19-s − 0.478·21-s − 0.563·23-s + 0.0598·25-s − 2.06·27-s + 0.138·29-s − 1.19·31-s + 0.810·33-s + 0.277·35-s + 0.612·37-s + 0.195·39-s − 1.54·41-s − 0.430·43-s + 2.22·45-s + 0.524·47-s − 0.927·49-s + 0.185·51-s − 0.264·53-s − 0.469·55-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 \( 1 + 3.07T + 3T^{2} \)
5 \( 1 - 2.30T + 5T^{2} \)
7 \( 1 - 0.712T + 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 + 0.395T + 13T^{2} \)
17 \( 1 + 0.429T + 17T^{2} \)
19 \( 1 - 3.54T + 19T^{2} \)
23 \( 1 + 2.70T + 23T^{2} \)
29 \( 1 - 0.745T + 29T^{2} \)
31 \( 1 + 6.64T + 31T^{2} \)
37 \( 1 - 3.72T + 37T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 + 2.82T + 43T^{2} \)
47 \( 1 - 3.59T + 47T^{2} \)
53 \( 1 + 1.92T + 53T^{2} \)
59 \( 1 + 1.45T + 59T^{2} \)
61 \( 1 - 1.00T + 61T^{2} \)
67 \( 1 + 2.83T + 67T^{2} \)
71 \( 1 + 5.09T + 71T^{2} \)
73 \( 1 + 3.35T + 73T^{2} \)
79 \( 1 - 1.70T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 - 5.06T + 89T^{2} \)
97 \( 1 - 3.13T + 97T^{2} \)
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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

−7.84779055618225338480501608140, −7.16972964576164032463409915342, −6.32990945622672505417980699379, −5.86216683541404148462415554157, −5.17613357469106661406379478331, −4.73229706965483246455712993719, −3.54518213860858286008835897737, −2.13715427569300537192239170735, −1.29975186323041556279854385929, 0, 1.29975186323041556279854385929, 2.13715427569300537192239170735, 3.54518213860858286008835897737, 4.73229706965483246455712993719, 5.17613357469106661406379478331, 5.86216683541404148462415554157, 6.32990945622672505417980699379, 7.16972964576164032463409915342, 7.84779055618225338480501608140

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