Properties

Label 4-70450-1.1-c1e2-0-2
Degree $4$
Conductor $70450$
Sign $-1$
Analytic cond. $4.49195$
Root an. cond. $1.45582$
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  − 2·2-s − 4·3-s + 4-s − 3·5-s + 8·6-s − 6·7-s + 2·8-s + 6·9-s + 6·10-s − 6·11-s − 4·12-s − 8·13-s + 12·14-s + 12·15-s − 3·16-s − 5·17-s − 12·18-s − 19-s − 3·20-s + 24·21-s + 12·22-s − 4·23-s − 8·24-s + 4·25-s + 16·26-s + 4·27-s − 6·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.34·5-s + 3.26·6-s − 2.26·7-s + 0.707·8-s + 2·9-s + 1.89·10-s − 1.80·11-s − 1.15·12-s − 2.21·13-s + 3.20·14-s + 3.09·15-s − 3/4·16-s − 1.21·17-s − 2.82·18-s − 0.229·19-s − 0.670·20-s + 5.23·21-s + 2.55·22-s − 0.834·23-s − 1.63·24-s + 4/5·25-s + 3.13·26-s + 0.769·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70450 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70450 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(70450\)    =    \(2 \cdot 5^{2} \cdot 1409\)
Sign: $-1$
Analytic conductor: \(4.49195\)
Root analytic conductor: \(1.45582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((4,\ 70450,\ (\ :1/2, 1/2),\ -1)\)

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_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
5$C_2$ \( 1 + 3 T + p T^{2} \)
1409$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 10 T + p T^{2} ) \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$D_{4}$ \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 7 T + 74 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T + 26 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T - 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 128 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$D_{4}$ \( 1 - T + 62 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
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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

−15.5879493094, −14.8924216832, −14.2170048351, −13.3968641966, −12.9847951448, −12.6095586737, −12.1770073882, −12.0031649433, −11.3680198006, −10.7666771035, −10.6026519496, −10.0676708642, −9.83024835264, −9.25824921610, −8.57564956157, −8.05245423278, −7.46267471926, −7.02418619054, −6.63048318511, −6.02574011821, −5.31594875322, −4.93266674710, −4.33287328926, −3.26336279818, −2.52080145024, 0, 0, 0, 2.52080145024, 3.26336279818, 4.33287328926, 4.93266674710, 5.31594875322, 6.02574011821, 6.63048318511, 7.02418619054, 7.46267471926, 8.05245423278, 8.57564956157, 9.25824921610, 9.83024835264, 10.0676708642, 10.6026519496, 10.7666771035, 11.3680198006, 12.0031649433, 12.1770073882, 12.6095586737, 12.9847951448, 13.3968641966, 14.2170048351, 14.8924216832, 15.5879493094

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