Properties

Label 2-4002-1.1-c1-0-32
Degree $2$
Conductor $4002$
Sign $1$
Analytic cond. $31.9561$
Root an. cond. $5.65297$
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-s − 3-s + 4-s + 3.59·5-s + 6-s + 1.23·7-s − 8-s + 9-s − 3.59·10-s + 0.260·11-s − 12-s + 5.91·13-s − 1.23·14-s − 3.59·15-s + 16-s − 4.05·17-s − 18-s + 6.94·19-s + 3.59·20-s − 1.23·21-s − 0.260·22-s + 23-s + 24-s + 7.94·25-s − 5.91·26-s − 27-s + 1.23·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.60·5-s + 0.408·6-s + 0.464·7-s − 0.353·8-s + 0.333·9-s − 1.13·10-s + 0.0784·11-s − 0.288·12-s + 1.64·13-s − 0.328·14-s − 0.929·15-s + 0.250·16-s − 0.984·17-s − 0.235·18-s + 1.59·19-s + 0.804·20-s − 0.268·21-s − 0.0554·22-s + 0.208·23-s + 0.204·24-s + 1.58·25-s − 1.16·26-s − 0.192·27-s + 0.232·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4002\)    =    \(2 \cdot 3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(31.9561\)
Root analytic conductor: \(5.65297\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.953065197\)
\(L(\frac12)\) \(\approx\) \(1.953065197\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
good5 \( 1 - 3.59T + 5T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 - 0.260T + 11T^{2} \)
13 \( 1 - 5.91T + 13T^{2} \)
17 \( 1 + 4.05T + 17T^{2} \)
19 \( 1 - 6.94T + 19T^{2} \)
31 \( 1 - 1.73T + 31T^{2} \)
37 \( 1 - 1.95T + 37T^{2} \)
41 \( 1 - 1.49T + 41T^{2} \)
43 \( 1 + 2.42T + 43T^{2} \)
47 \( 1 - 2.76T + 47T^{2} \)
53 \( 1 + 2.88T + 53T^{2} \)
59 \( 1 - 0.426T + 59T^{2} \)
61 \( 1 + 5.37T + 61T^{2} \)
67 \( 1 - 15.5T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 1.07T + 73T^{2} \)
79 \( 1 - 5.02T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−8.546230897692057138533108236243, −7.83330739329757817726815651296, −6.70340990599368591092550559535, −6.39604839453110415759367122178, −5.56173003943525570699708667975, −5.05008325526529854422952873931, −3.79117649733069022940862403101, −2.63980689308710913991800128661, −1.64627094991400125478324745277, −1.02653446190447595663648990950, 1.02653446190447595663648990950, 1.64627094991400125478324745277, 2.63980689308710913991800128661, 3.79117649733069022940862403101, 5.05008325526529854422952873931, 5.56173003943525570699708667975, 6.39604839453110415759367122178, 6.70340990599368591092550559535, 7.83330739329757817726815651296, 8.546230897692057138533108236243

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