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Properties

Label 4-1568-1.1-c1e2-0-1
Degree $4$
Conductor $1568$
Sign $1$
Analytic cond. $0.0999770$
Root an. cond. $0.562309$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 2·9-s − 8·13-s + 16-s + 12·17-s + 2·18-s − 10·25-s + 8·26-s − 12·29-s − 32-s − 12·34-s − 2·36-s + 4·37-s + 12·41-s + 49-s + 10·50-s − 8·52-s + 12·53-s + 12·58-s + 16·61-s + 64-s + 12·68-s + 2·72-s + 4·73-s − 4·74-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s − 2.21·13-s + 1/4·16-s + 2.91·17-s + 0.471·18-s − 2·25-s + 1.56·26-s − 2.22·29-s − 0.176·32-s − 2.05·34-s − 1/3·36-s + 0.657·37-s + 1.87·41-s + 1/7·49-s + 1.41·50-s − 1.10·52-s + 1.64·53-s + 1.57·58-s + 2.04·61-s + 1/8·64-s + 1.45·68-s + 0.235·72-s + 0.468·73-s − 0.464·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.0999770\)
Root analytic conductor: \(0.562309\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1568,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4377085675\)
\(L(\frac12)\) \(\approx\) \(0.4377085675\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−13.78118158860551012757665837778, −12.74339932248381599513991172244, −12.30526099005271094386573698884, −11.71064761043096959517781978032, −11.23136141438460806904855801814, −10.11465006559287644470603673937, −9.765547119459919407856461234632, −9.392494157732895222790142944171, −8.182643116839158569055135937831, −7.57571100088867902110310233811, −7.29668061464756137953129595214, −5.62168100149868099368912779374, −5.57928681742950427486583645839, −3.82418745638366191622617161116, −2.48514181668809031419660323366, 2.48514181668809031419660323366, 3.82418745638366191622617161116, 5.57928681742950427486583645839, 5.62168100149868099368912779374, 7.29668061464756137953129595214, 7.57571100088867902110310233811, 8.182643116839158569055135937831, 9.392494157732895222790142944171, 9.765547119459919407856461234632, 10.11465006559287644470603673937, 11.23136141438460806904855801814, 11.71064761043096959517781978032, 12.30526099005271094386573698884, 12.74339932248381599513991172244, 13.78118158860551012757665837778

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