Properties

Label 4-100283-1.1-c1e2-0-0
Degree $4$
Conductor $100283$
Sign $1$
Analytic cond. $6.39413$
Root an. cond. $1.59017$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 3·5-s + 6-s − 3·8-s + 3·10-s − 5·11-s − 12-s − 7·13-s + 3·15-s + 16-s − 2·17-s − 4·19-s − 3·20-s + 5·22-s − 4·23-s + 3·24-s + 5·25-s + 7·26-s − 2·27-s + 3·29-s − 3·30-s − 11·31-s + 32-s + 5·33-s + 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.06·8-s + 0.948·10-s − 1.50·11-s − 0.288·12-s − 1.94·13-s + 0.774·15-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.670·20-s + 1.06·22-s − 0.834·23-s + 0.612·24-s + 25-s + 1.37·26-s − 0.384·27-s + 0.557·29-s − 0.547·30-s − 1.97·31-s + 0.176·32-s + 0.870·33-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100283 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100283 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100283\)    =    \(17^{2} \cdot 347\)
Sign: $1$
Analytic conductor: \(6.39413\)
Root analytic conductor: \(1.59017\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100283,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad17$C_2$ \( 1 + 2 T + p T^{2} \)
347$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 15 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 36 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 11 T + 71 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$D_{4}$ \( 1 + T - p T^{2} + p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T - 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T + 112 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
89$D_{4}$ \( 1 - 11 T + 123 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−14.6987439589, −14.2930123098, −13.4227757441, −12.9997420247, −12.5833609762, −12.0916652298, −11.9364202458, −11.3597309402, −11.0014391413, −10.4971364404, −10.1600942090, −9.53804732924, −9.07203478161, −8.56398584444, −7.91714752584, −7.70457902094, −7.25095688486, −6.64583170125, −6.13811607972, −5.27751949503, −5.02600758989, −4.25549372233, −3.58529418283, −2.59515781847, −2.25380056961, 0, 0, 2.25380056961, 2.59515781847, 3.58529418283, 4.25549372233, 5.02600758989, 5.27751949503, 6.13811607972, 6.64583170125, 7.25095688486, 7.70457902094, 7.91714752584, 8.56398584444, 9.07203478161, 9.53804732924, 10.1600942090, 10.4971364404, 11.0014391413, 11.3597309402, 11.9364202458, 12.0916652298, 12.5833609762, 12.9997420247, 13.4227757441, 14.2930123098, 14.6987439589

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