Properties

Label 2-4002-1.1-c1-0-19
Degree $2$
Conductor $4002$
Sign $1$
Analytic cond. $31.9561$
Root an. cond. $5.65297$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·5-s + 6-s − 4·7-s + 8-s + 9-s − 2·10-s + 12-s − 2·13-s − 4·14-s − 2·15-s + 16-s + 6·17-s + 18-s + 4·19-s − 2·20-s − 4·21-s − 23-s + 24-s − 25-s − 2·26-s + 27-s − 4·28-s + 29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.872·21-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 0.185·29-s − 0.365·30-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.582431080\)
\(L(\frac12)\) \(\approx\) \(2.582431080\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.117923352354560842576493810952, −7.72241330075962692237819063696, −7.01333323419057346375191818255, −6.24812917471354955721910234661, −5.46356916712438031826540105298, −4.50162287690062263138037116263, −3.55784116503777040358801542907, −3.29995376479477247004032977433, −2.38745920662164929006825210076, −0.792012704452018534737806150565, 0.792012704452018534737806150565, 2.38745920662164929006825210076, 3.29995376479477247004032977433, 3.55784116503777040358801542907, 4.50162287690062263138037116263, 5.46356916712438031826540105298, 6.24812917471354955721910234661, 7.01333323419057346375191818255, 7.72241330075962692237819063696, 8.117923352354560842576493810952

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