Properties

Label 4-10023-1.1-c1e2-0-0
Degree $4$
Conductor $10023$
Sign $-1$
Analytic cond. $0.639075$
Root an. cond. $0.894103$
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  − 2·2-s + 2·4-s + 5-s − 7-s − 4·8-s − 2·9-s − 2·10-s − 6·11-s + 2·14-s + 8·16-s − 3·17-s + 4·18-s − 3·19-s + 2·20-s + 12·22-s − 3·23-s − 2·25-s − 3·27-s − 2·28-s − 3·29-s − 8·32-s + 6·34-s − 35-s − 4·36-s + 10·37-s + 6·38-s − 4·40-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 1.41·8-s − 2/3·9-s − 0.632·10-s − 1.80·11-s + 0.534·14-s + 2·16-s − 0.727·17-s + 0.942·18-s − 0.688·19-s + 0.447·20-s + 2.55·22-s − 0.625·23-s − 2/5·25-s − 0.577·27-s − 0.377·28-s − 0.557·29-s − 1.41·32-s + 1.02·34-s − 0.169·35-s − 2/3·36-s + 1.64·37-s + 0.973·38-s − 0.632·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10023 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10023 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10023\)    =    \(3 \cdot 13 \cdot 257\)
Sign: $-1$
Analytic conductor: \(0.639075\)
Root analytic conductor: \(0.894103\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10023} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 10023,\ (\ :1/2, 1/2),\ -1)\)

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$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
257$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 24 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$D_{4}$ \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T - 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$D_{4}$ \( 1 - T + 31 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 5 T + 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 7 T + 135 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 145 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.9315675446, −16.4103256545, −16.0611751297, −15.4102538548, −15.0152858908, −14.6487150518, −13.7251714102, −13.2674183342, −12.8971819445, −12.2149951824, −11.6369800362, −10.9789253317, −10.6000656074, −9.90791884600, −9.64650193093, −8.87851560574, −8.61161327080, −7.79606558193, −7.55809667704, −6.42905021193, −5.89625010754, −5.46682937590, −4.20503897888, −2.90536555045, −2.31894101908, 0, 2.31894101908, 2.90536555045, 4.20503897888, 5.46682937590, 5.89625010754, 6.42905021193, 7.55809667704, 7.79606558193, 8.61161327080, 8.87851560574, 9.64650193093, 9.90791884600, 10.6000656074, 10.9789253317, 11.6369800362, 12.2149951824, 12.8971819445, 13.2674183342, 13.7251714102, 14.6487150518, 15.0152858908, 15.4102538548, 16.0611751297, 16.4103256545, 16.9315675446

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