Properties

Label 2-4002-1.1-c1-0-27
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 2.88·5-s + 6-s − 3.71·7-s − 8-s + 9-s + 2.88·10-s + 5.50·11-s − 12-s − 3.61·13-s + 3.71·14-s + 2.88·15-s + 16-s − 4.95·17-s − 18-s + 4.95·19-s − 2.88·20-s + 3.71·21-s − 5.50·22-s − 23-s + 24-s + 3.33·25-s + 3.61·26-s − 27-s − 3.71·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.29·5-s + 0.408·6-s − 1.40·7-s − 0.353·8-s + 0.333·9-s + 0.913·10-s + 1.66·11-s − 0.288·12-s − 1.00·13-s + 0.993·14-s + 0.745·15-s + 0.250·16-s − 1.20·17-s − 0.235·18-s + 1.13·19-s − 0.645·20-s + 0.810·21-s − 1.17·22-s − 0.208·23-s + 0.204·24-s + 0.667·25-s + 0.709·26-s − 0.192·27-s − 0.702·28-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: \(1\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.88T + 5T^{2} \)
7 \( 1 + 3.71T + 7T^{2} \)
11 \( 1 - 5.50T + 11T^{2} \)
13 \( 1 + 3.61T + 13T^{2} \)
17 \( 1 + 4.95T + 17T^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
31 \( 1 - 6.02T + 31T^{2} \)
37 \( 1 + 1.34T + 37T^{2} \)
41 \( 1 + 1.77T + 41T^{2} \)
43 \( 1 + 7.39T + 43T^{2} \)
47 \( 1 - 13.6T + 47T^{2} \)
53 \( 1 - 2.70T + 53T^{2} \)
59 \( 1 - 8.87T + 59T^{2} \)
61 \( 1 + 5.99T + 61T^{2} \)
67 \( 1 - 9.67T + 67T^{2} \)
71 \( 1 - 3.39T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 4.99T + 79T^{2} \)
83 \( 1 - 9.98T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + 6.70T + 97T^{2} \)
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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

−8.056733426744796758800390779368, −7.16834339684422802381238351910, −6.80036613043399725917409759551, −6.23292262448878159128716463037, −5.07698801855859205716715204747, −4.04358760341228978143757835957, −3.55835338934226451363465026818, −2.44988023498937485379709386146, −0.937170139549211323205395888021, 0, 0.937170139549211323205395888021, 2.44988023498937485379709386146, 3.55835338934226451363465026818, 4.04358760341228978143757835957, 5.07698801855859205716715204747, 6.23292262448878159128716463037, 6.80036613043399725917409759551, 7.16834339684422802381238351910, 8.056733426744796758800390779368

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