Properties

Label 2-4016-1.1-c1-0-22
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  − 1.66·3-s − 3.99·5-s + 2.26·7-s − 0.211·9-s + 4.12·11-s + 5.91·13-s + 6.67·15-s − 1.43·17-s − 0.204·19-s − 3.77·21-s − 6.71·23-s + 10.9·25-s + 5.36·27-s − 1.38·29-s + 3.06·31-s − 6.88·33-s − 9.04·35-s + 4.37·37-s − 9.87·39-s − 7.17·41-s − 9.92·43-s + 0.845·45-s − 6.19·47-s − 1.88·49-s + 2.39·51-s + 2.85·53-s − 16.4·55-s + ⋯
L(s)  = 1  − 0.964·3-s − 1.78·5-s + 0.855·7-s − 0.0705·9-s + 1.24·11-s + 1.63·13-s + 1.72·15-s − 0.347·17-s − 0.0469·19-s − 0.824·21-s − 1.40·23-s + 2.19·25-s + 1.03·27-s − 0.257·29-s + 0.550·31-s − 1.19·33-s − 1.52·35-s + 0.719·37-s − 1.58·39-s − 1.12·41-s − 1.51·43-s + 0.126·45-s − 0.904·47-s − 0.268·49-s + 0.335·51-s + 0.392·53-s − 2.22·55-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

\(L(1)\) \(\approx\) \(0.9304161939\)
\(L(\frac12)\) \(\approx\) \(0.9304161939\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.66T + 3T^{2} \)
5 \( 1 + 3.99T + 5T^{2} \)
7 \( 1 - 2.26T + 7T^{2} \)
11 \( 1 - 4.12T + 11T^{2} \)
13 \( 1 - 5.91T + 13T^{2} \)
17 \( 1 + 1.43T + 17T^{2} \)
19 \( 1 + 0.204T + 19T^{2} \)
23 \( 1 + 6.71T + 23T^{2} \)
29 \( 1 + 1.38T + 29T^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 - 4.37T + 37T^{2} \)
41 \( 1 + 7.17T + 41T^{2} \)
43 \( 1 + 9.92T + 43T^{2} \)
47 \( 1 + 6.19T + 47T^{2} \)
53 \( 1 - 2.85T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 1.35T + 61T^{2} \)
67 \( 1 - 4.56T + 67T^{2} \)
71 \( 1 + 0.472T + 71T^{2} \)
73 \( 1 + 3.38T + 73T^{2} \)
79 \( 1 - 4.84T + 79T^{2} \)
83 \( 1 - 4.30T + 83T^{2} \)
89 \( 1 - 2.35T + 89T^{2} \)
97 \( 1 + 6.51T + 97T^{2} \)
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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

−8.345049663687800610213160024525, −7.920879410270889587256128834201, −6.74554382521507578476793298633, −6.43179861365777412179026812828, −5.43156166071416913889806837581, −4.54366707644214240405277436918, −3.97199763132510248690207460899, −3.33481396366813312843539383394, −1.62087326959790614701491510137, −0.61214022298307242884467387285, 0.61214022298307242884467387285, 1.62087326959790614701491510137, 3.33481396366813312843539383394, 3.97199763132510248690207460899, 4.54366707644214240405277436918, 5.43156166071416913889806837581, 6.43179861365777412179026812828, 6.74554382521507578476793298633, 7.920879410270889587256128834201, 8.345049663687800610213160024525

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