Properties

Label 4-164e2-1.1-c1e2-0-4
Degree $4$
Conductor $26896$
Sign $1$
Analytic cond. $1.71491$
Root an. cond. $1.14435$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 4·5-s + 2·6-s − 6·7-s − 8-s − 2·9-s + 4·10-s − 2·11-s − 2·12-s + 6·14-s + 8·15-s + 16-s − 4·17-s + 2·18-s + 10·19-s − 4·20-s + 12·21-s + 2·22-s − 12·23-s + 2·24-s + 2·25-s + 10·27-s − 6·28-s − 8·30-s − 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s − 2.26·7-s − 0.353·8-s − 2/3·9-s + 1.26·10-s − 0.603·11-s − 0.577·12-s + 1.60·14-s + 2.06·15-s + 1/4·16-s − 0.970·17-s + 0.471·18-s + 2.29·19-s − 0.894·20-s + 2.61·21-s + 0.426·22-s − 2.50·23-s + 0.408·24-s + 2/5·25-s + 1.92·27-s − 1.13·28-s − 1.46·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26896\)    =    \(2^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(1.71491\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 26896,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( 1 + T \)
41$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
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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

−16.0384623536, −15.6789109424, −15.5438761265, −14.5545960804, −13.9030213936, −13.5603503236, −12.7499074846, −12.3535797825, −11.9406123124, −11.5374061458, −11.3637312458, −10.5945207017, −10.1450105097, −9.48966114694, −9.29245441693, −8.21258579084, −8.02435745103, −7.37965720962, −6.70748925180, −6.27411578104, −5.69856539644, −5.09110565346, −3.78065861132, −3.54075497824, −2.66249098499, 0, 0, 2.66249098499, 3.54075497824, 3.78065861132, 5.09110565346, 5.69856539644, 6.27411578104, 6.70748925180, 7.37965720962, 8.02435745103, 8.21258579084, 9.29245441693, 9.48966114694, 10.1450105097, 10.5945207017, 11.3637312458, 11.5374061458, 11.9406123124, 12.3535797825, 12.7499074846, 13.5603503236, 13.9030213936, 14.5545960804, 15.5438761265, 15.6789109424, 16.0384623536

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