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Properties

Label 2-10002-1.1-c1-0-3
Degree $2$
Conductor $10002$
Sign $-1$
Analytic cond. $79.8663$
Root an. cond. $8.93679$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 2·5-s + 6-s + 4·7-s − 8-s + 9-s − 2·10-s − 4·11-s − 12-s + 2·13-s − 4·14-s − 2·15-s + 16-s − 3·17-s − 18-s − 4·19-s + 2·20-s − 4·21-s + 4·22-s − 3·23-s + 24-s − 25-s − 2·26-s − 27-s + 4·28-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 − 1.20·11-s − 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.872·21-s + 0.852·22-s − 0.625·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10002\)    =    \(2 \cdot 3 \cdot 1667\)
Sign: $-1$
Analytic conductor: \(79.8663\)
Root analytic conductor: \(8.93679\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10002,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
1667 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−17.01201475492935, −16.65543122014959, −15.74644476994452, −15.36511979058054, −14.75152809318323, −14.07628203498126, −13.28932380177414, −13.07580179930266, −12.11972739635648, −11.42992092825958, −10.99905286556535, −10.58536749292370, −9.954857409808653, −9.287162990292037, −8.533066932894288, −7.980642471441895, −7.536445547256014, −6.566835392655653, −5.933880611988324, −5.439547892125127, −4.687769515237060, −3.975887338217949, −2.507707004237967, −2.040497678549198, −1.268010831947743, 0, 1.268010831947743, 2.040497678549198, 2.507707004237967, 3.975887338217949, 4.687769515237060, 5.439547892125127, 5.933880611988324, 6.566835392655653, 7.536445547256014, 7.980642471441895, 8.533066932894288, 9.287162990292037, 9.954857409808653, 10.58536749292370, 10.99905286556535, 11.42992092825958, 12.11972739635648, 13.07580179930266, 13.28932380177414, 14.07628203498126, 14.75152809318323, 15.36511979058054, 15.74644476994452, 16.65543122014959, 17.01201475492935

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