Properties

Label 2-4016-1.1-c1-0-58
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  + 2.79·3-s − 3.17·5-s + 1.46·7-s + 4.79·9-s + 4.08·11-s + 4.91·13-s − 8.85·15-s + 3.89·17-s + 4.08·19-s + 4.09·21-s − 7.08·23-s + 5.05·25-s + 4.99·27-s − 1.48·29-s − 9.24·31-s + 11.3·33-s − 4.65·35-s + 2.10·37-s + 13.7·39-s − 6.48·41-s + 5.95·43-s − 15.1·45-s + 8.02·47-s − 4.84·49-s + 10.8·51-s + 4.69·53-s − 12.9·55-s + ⋯
L(s)  = 1  + 1.61·3-s − 1.41·5-s + 0.554·7-s + 1.59·9-s + 1.23·11-s + 1.36·13-s − 2.28·15-s + 0.944·17-s + 0.937·19-s + 0.894·21-s − 1.47·23-s + 1.01·25-s + 0.961·27-s − 0.275·29-s − 1.66·31-s + 1.98·33-s − 0.787·35-s + 0.346·37-s + 2.19·39-s − 1.01·41-s + 0.907·43-s − 2.26·45-s + 1.17·47-s − 0.692·49-s + 1.52·51-s + 0.644·53-s − 1.74·55-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

\(L(1)\) \(\approx\) \(3.449902055\)
\(L(\frac12)\) \(\approx\) \(3.449902055\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.79T + 3T^{2} \)
5 \( 1 + 3.17T + 5T^{2} \)
7 \( 1 - 1.46T + 7T^{2} \)
11 \( 1 - 4.08T + 11T^{2} \)
13 \( 1 - 4.91T + 13T^{2} \)
17 \( 1 - 3.89T + 17T^{2} \)
19 \( 1 - 4.08T + 19T^{2} \)
23 \( 1 + 7.08T + 23T^{2} \)
29 \( 1 + 1.48T + 29T^{2} \)
31 \( 1 + 9.24T + 31T^{2} \)
37 \( 1 - 2.10T + 37T^{2} \)
41 \( 1 + 6.48T + 41T^{2} \)
43 \( 1 - 5.95T + 43T^{2} \)
47 \( 1 - 8.02T + 47T^{2} \)
53 \( 1 - 4.69T + 53T^{2} \)
59 \( 1 + 5.12T + 59T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 - 1.10T + 67T^{2} \)
71 \( 1 + 1.84T + 71T^{2} \)
73 \( 1 - 3.34T + 73T^{2} \)
79 \( 1 + 2.16T + 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 - 4.28T + 97T^{2} \)
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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

−8.408652489691230921703202122132, −7.70101615844144170617375566252, −7.52322166657194417632312742488, −6.43748491656366111427431455342, −5.38098867675891907779061582759, −4.05348945794364889199936620426, −3.80960071729518161985854086023, −3.31333953403120909759337517554, −1.98330409474765685023284193907, −1.07799568295391165647600602240, 1.07799568295391165647600602240, 1.98330409474765685023284193907, 3.31333953403120909759337517554, 3.80960071729518161985854086023, 4.05348945794364889199936620426, 5.38098867675891907779061582759, 6.43748491656366111427431455342, 7.52322166657194417632312742488, 7.70101615844144170617375566252, 8.408652489691230921703202122132

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