Properties

Label 2-4016-1.1-c1-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.505·3-s − 3.97·5-s − 1.36·7-s − 2.74·9-s − 4.72·11-s − 3.07·13-s − 2.00·15-s − 2.19·17-s − 5.12·19-s − 0.690·21-s − 5.63·23-s + 10.8·25-s − 2.90·27-s + 8.38·29-s − 5.97·31-s − 2.38·33-s + 5.43·35-s − 0.0197·37-s − 1.55·39-s − 8.14·41-s − 1.90·43-s + 10.9·45-s + 1.09·47-s − 5.12·49-s − 1.10·51-s + 7.79·53-s + 18.7·55-s + ⋯
L(s)  = 1  + 0.291·3-s − 1.77·5-s − 0.516·7-s − 0.914·9-s − 1.42·11-s − 0.852·13-s − 0.518·15-s − 0.532·17-s − 1.17·19-s − 0.150·21-s − 1.17·23-s + 2.16·25-s − 0.558·27-s + 1.55·29-s − 1.07·31-s − 0.415·33-s + 0.919·35-s − 0.00324·37-s − 0.248·39-s − 1.27·41-s − 0.290·43-s + 1.62·45-s + 0.159·47-s − 0.732·49-s − 0.155·51-s + 1.07·53-s + 2.53·55-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: \(0\)
Selberg data: \((2,\ 4016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05612054933\)
\(L(\frac12)\) \(\approx\) \(0.05612054933\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.505T + 3T^{2} \)
5 \( 1 + 3.97T + 5T^{2} \)
7 \( 1 + 1.36T + 7T^{2} \)
11 \( 1 + 4.72T + 11T^{2} \)
13 \( 1 + 3.07T + 13T^{2} \)
17 \( 1 + 2.19T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
23 \( 1 + 5.63T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 + 5.97T + 31T^{2} \)
37 \( 1 + 0.0197T + 37T^{2} \)
41 \( 1 + 8.14T + 41T^{2} \)
43 \( 1 + 1.90T + 43T^{2} \)
47 \( 1 - 1.09T + 47T^{2} \)
53 \( 1 - 7.79T + 53T^{2} \)
59 \( 1 + 1.16T + 59T^{2} \)
61 \( 1 - 7.58T + 61T^{2} \)
67 \( 1 - 7.25T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 - 4.79T + 79T^{2} \)
83 \( 1 - 3.89T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 0.906T + 97T^{2} \)
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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

−8.425432386831450755313704665163, −7.82418762462146709845279703410, −7.16901315802953909642439733851, −6.37588658006153263683995379176, −5.32241208597903735350302659552, −4.57624672898882976090056475311, −3.79618790858218832729083586028, −2.97548235291208678613950461731, −2.31359206522672441955347062819, −0.12136815475497691553484153397, 0.12136815475497691553484153397, 2.31359206522672441955347062819, 2.97548235291208678613950461731, 3.79618790858218832729083586028, 4.57624672898882976090056475311, 5.32241208597903735350302659552, 6.37588658006153263683995379176, 7.16901315802953909642439733851, 7.82418762462146709845279703410, 8.425432386831450755313704665163

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