Properties

Label 4-438869-1.1-c1e2-0-0
Degree $4$
Conductor $438869$
Sign $-1$
Analytic cond. $27.9826$
Root an. cond. $2.29997$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 4-s − 3·5-s + 3·6-s − 2·7-s + 8-s + 4·9-s + 3·10-s − 6·11-s + 3·12-s − 3·13-s + 2·14-s + 9·15-s − 16-s − 2·17-s − 4·18-s − 8·19-s + 3·20-s + 6·21-s + 6·22-s − 2·23-s − 3·24-s + 3·26-s − 6·27-s + 2·28-s − 8·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 1/2·4-s − 1.34·5-s + 1.22·6-s − 0.755·7-s + 0.353·8-s + 4/3·9-s + 0.948·10-s − 1.80·11-s + 0.866·12-s − 0.832·13-s + 0.534·14-s + 2.32·15-s − 1/4·16-s − 0.485·17-s − 0.942·18-s − 1.83·19-s + 0.670·20-s + 1.30·21-s + 1.27·22-s − 0.417·23-s − 0.612·24-s + 0.588·26-s − 1.15·27-s + 0.377·28-s − 1.48·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 438869 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 438869 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(438869\)
Sign: $-1$
Analytic conductor: \(27.9826\)
Root analytic conductor: \(2.29997\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{438869} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((4,\ 438869,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad438869$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 515 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$D_{4}$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 13 T + 118 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T - 94 T^{2} + p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 3 T + 125 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
89$D_{4}$ \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 15 T + 130 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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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

−13.2717994866, −12.8477676366, −12.3588149021, −12.2293850522, −11.6336710723, −11.1865344764, −11.0175972351, −10.6315682964, −10.0892159299, −9.74548935752, −9.43467096563, −8.68742504409, −8.30430360917, −7.95542741678, −7.52952174514, −6.98500271793, −6.50905463274, −6.12308457194, −5.39765796460, −5.14527260024, −4.66170111454, −4.00624531870, −3.64909755483, −2.66493687136, −1.93831896812, 0, 0, 0, 1.93831896812, 2.66493687136, 3.64909755483, 4.00624531870, 4.66170111454, 5.14527260024, 5.39765796460, 6.12308457194, 6.50905463274, 6.98500271793, 7.52952174514, 7.95542741678, 8.30430360917, 8.68742504409, 9.43467096563, 9.74548935752, 10.0892159299, 10.6315682964, 11.0175972351, 11.1865344764, 11.6336710723, 12.2293850522, 12.3588149021, 12.8477676366, 13.2717994866

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