Properties

Label 2-4016-1.1-c1-0-46
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  − 1.99·3-s + 2.61·5-s + 4.58·7-s + 0.968·9-s + 4.22·11-s − 4.74·13-s − 5.21·15-s − 3.73·17-s − 3.23·19-s − 9.13·21-s + 6.76·23-s + 1.84·25-s + 4.04·27-s + 7.19·29-s + 1.31·31-s − 8.40·33-s + 11.9·35-s − 3.53·37-s + 9.45·39-s + 7.91·41-s − 2.99·43-s + 2.53·45-s + 4.75·47-s + 14.0·49-s + 7.44·51-s + 2.40·53-s + 11.0·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.16·5-s + 1.73·7-s + 0.322·9-s + 1.27·11-s − 1.31·13-s − 1.34·15-s − 0.906·17-s − 0.743·19-s − 1.99·21-s + 1.41·23-s + 0.368·25-s + 0.778·27-s + 1.33·29-s + 0.236·31-s − 1.46·33-s + 2.02·35-s − 0.580·37-s + 1.51·39-s + 1.23·41-s − 0.457·43-s + 0.377·45-s + 0.693·47-s + 2.00·49-s + 1.04·51-s + 0.330·53-s + 1.48·55-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

\(L(1)\) \(\approx\) \(2.011175851\)
\(L(\frac12)\) \(\approx\) \(2.011175851\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.99T + 3T^{2} \)
5 \( 1 - 2.61T + 5T^{2} \)
7 \( 1 - 4.58T + 7T^{2} \)
11 \( 1 - 4.22T + 11T^{2} \)
13 \( 1 + 4.74T + 13T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 + 3.23T + 19T^{2} \)
23 \( 1 - 6.76T + 23T^{2} \)
29 \( 1 - 7.19T + 29T^{2} \)
31 \( 1 - 1.31T + 31T^{2} \)
37 \( 1 + 3.53T + 37T^{2} \)
41 \( 1 - 7.91T + 41T^{2} \)
43 \( 1 + 2.99T + 43T^{2} \)
47 \( 1 - 4.75T + 47T^{2} \)
53 \( 1 - 2.40T + 53T^{2} \)
59 \( 1 - 7.15T + 59T^{2} \)
61 \( 1 + 8.22T + 61T^{2} \)
67 \( 1 + 0.0714T + 67T^{2} \)
71 \( 1 - 3.10T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 4.94T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 - 18.2T + 89T^{2} \)
97 \( 1 + 2.58T + 97T^{2} \)
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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

−8.763998169279552903597368577442, −7.53315030035024634820718286650, −6.78446900850313920567543496963, −6.20355958908122274623734542427, −5.40858836514045842396806100895, −4.77471052726112560881050514630, −4.37746889624423805927029796557, −2.62372931569775348681200798382, −1.82202397533453168977522798395, −0.912438228978432677554837170099, 0.912438228978432677554837170099, 1.82202397533453168977522798395, 2.62372931569775348681200798382, 4.37746889624423805927029796557, 4.77471052726112560881050514630, 5.40858836514045842396806100895, 6.20355958908122274623734542427, 6.78446900850313920567543496963, 7.53315030035024634820718286650, 8.763998169279552903597368577442

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