Properties

Label 4-100227-1.1-c1e2-0-0
Degree $4$
Conductor $100227$
Sign $-1$
Analytic cond. $6.39056$
Root an. cond. $1.58995$
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 + 2·7-s + 4·8-s − 2·9-s − 2·11-s + 7·13-s − 4·14-s − 4·16-s − 3·17-s + 4·18-s + 19-s + 4·22-s − 14·26-s + 3·27-s − 6·29-s − 10·31-s + 6·34-s − 8·37-s − 2·38-s − 41-s − 4·43-s + 47-s − 6·54-s + 8·56-s + 12·58-s − 2·59-s − 14·61-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.755·7-s + 1.41·8-s − 2/3·9-s − 0.603·11-s + 1.94·13-s − 1.06·14-s − 16-s − 0.727·17-s + 0.942·18-s + 0.229·19-s + 0.852·22-s − 2.74·26-s + 0.577·27-s − 1.11·29-s − 1.79·31-s + 1.02·34-s − 1.31·37-s − 0.324·38-s − 0.156·41-s − 0.609·43-s + 0.145·47-s − 0.816·54-s + 1.06·56-s + 1.57·58-s − 0.260·59-s − 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100227 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100227 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100227\)    =    \(3 \cdot 33409\)
Sign: $-1$
Analytic conductor: \(6.39056\)
Root analytic conductor: \(1.58995\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 100227,\ (\ :1/2, 1/2),\ -1)\)

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$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
33409$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 126 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 7 T + 29 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 64 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$D_{4}$ \( 1 + T - 49 T^{2} + p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 16 T + 170 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 5 T - 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 19 T + 189 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 15 T + 132 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + T + 94 T^{2} + p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−14.0865882068, −13.7708105309, −13.5255311119, −13.1208915989, −12.4611031786, −12.0326113961, −11.2413018547, −11.0273477341, −10.6831347731, −10.3982266581, −9.50973101555, −9.14266189535, −8.93565428390, −8.40007668318, −8.07013014232, −7.69349833373, −7.00414335712, −6.35310953240, −5.69723724538, −5.18592186603, −4.63859595986, −3.79220369006, −3.31916970588, −2.04535531759, −1.32355243103, 0, 1.32355243103, 2.04535531759, 3.31916970588, 3.79220369006, 4.63859595986, 5.18592186603, 5.69723724538, 6.35310953240, 7.00414335712, 7.69349833373, 8.07013014232, 8.40007668318, 8.93565428390, 9.14266189535, 9.50973101555, 10.3982266581, 10.6831347731, 11.0273477341, 11.2413018547, 12.0326113961, 12.4611031786, 13.1208915989, 13.5255311119, 13.7708105309, 14.0865882068

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