Properties

Label 4-10056-1.1-c1e2-0-0
Degree $4$
Conductor $10056$
Sign $-1$
Analytic cond. $0.641179$
Root an. cond. $0.894838$
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 − 5·7-s + 3·8-s − 2·10-s − 11-s + 12-s − 6·13-s + 5·14-s − 2·15-s − 16-s + 3·17-s − 6·19-s − 2·20-s + 5·21-s + 22-s − 3·24-s − 4·25-s + 6·26-s + 4·27-s + 5·28-s + 4·29-s + 2·30-s − 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.88·7-s + 1.06·8-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 1.66·13-s + 1.33·14-s − 0.516·15-s − 1/4·16-s + 0.727·17-s − 1.37·19-s − 0.447·20-s + 1.09·21-s + 0.213·22-s − 0.612·24-s − 4/5·25-s + 1.17·26-s + 0.769·27-s + 0.944·28-s + 0.742·29-s + 0.365·30-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10056\)    =    \(2^{3} \cdot 3 \cdot 419\)
Sign: $-1$
Analytic conductor: \(0.641179\)
Root analytic conductor: \(0.894838\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 10056,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_2$ \( 1 + T + p T^{2} \)
3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
419$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 20 T + p T^{2} ) \)
good5$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$D_{4}$ \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + T - 24 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T - 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T - 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 7 T + 50 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 3 T - 120 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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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

−16.9221529890, −16.6457327236, −16.0575548591, −15.5592650392, −14.7949917622, −14.3371538918, −13.7936150994, −13.1999770747, −12.8447732949, −12.3791637204, −12.0051872293, −10.9118083646, −10.3774437183, −10.0745691727, −9.61858958358, −9.23133480024, −8.58936996280, −7.69512942194, −7.14785607019, −6.43414449648, −5.87342079481, −5.20327237273, −4.39691775739, −3.31113675149, −2.20975427653, 0, 2.20975427653, 3.31113675149, 4.39691775739, 5.20327237273, 5.87342079481, 6.43414449648, 7.14785607019, 7.69512942194, 8.58936996280, 9.23133480024, 9.61858958358, 10.0745691727, 10.3774437183, 10.9118083646, 12.0051872293, 12.3791637204, 12.8447732949, 13.1999770747, 13.7936150994, 14.3371538918, 14.7949917622, 15.5592650392, 16.0575548591, 16.6457327236, 16.9221529890

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