Properties

Degree $2$
Conductor $4002$
Sign $1$
Motivic weight $1$
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 − 5-s − 6-s − 8-s + 9-s + 10-s + 5·11-s + 12-s + 3·13-s − 15-s + 16-s + 4·17-s − 18-s + 4·19-s − 20-s − 5·22-s − 23-s − 24-s − 4·25-s − 3·26-s + 27-s − 29-s + 30-s + 5·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s + 0.288·12-s + 0.832·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 1.06·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.588·26-s + 0.192·27-s − 0.185·29-s + 0.182·30-s + 0.898·31-s − 0.176·32-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: $1$
Motivic weight: \(1\)
Character: $\chi_{4002} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.958439422\)
\(L(\frac12)\) \(\approx\) \(1.958439422\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−18.18419869261203, −17.54042504755313, −16.89356118077553, −16.27837835037836, −15.73040523187947, −15.15553508957050, −14.33893821977068, −14.00086089502371, −13.20293436441238, −12.20094966503532, −11.81237000364181, −11.25610863296316, −10.29069619234508, −9.748327985448738, −9.025200701122200, −8.582903535103052, −7.701076747858817, −7.317155684768198, −6.328910827474636, −5.749859363461188, −4.419558746858169, −3.676596612698982, −3.035381561050612, −1.709654497512743, −0.9460944564048167, 0.9460944564048167, 1.709654497512743, 3.035381561050612, 3.676596612698982, 4.419558746858169, 5.749859363461188, 6.328910827474636, 7.317155684768198, 7.701076747858817, 8.582903535103052, 9.025200701122200, 9.748327985448738, 10.29069619234508, 11.25610863296316, 11.81237000364181, 12.20094966503532, 13.20293436441238, 14.00086089502371, 14.33893821977068, 15.15553508957050, 15.73040523187947, 16.27837835037836, 16.89356118077553, 17.54042504755313, 18.18419869261203

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