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Properties

Label 4-1576e2-1.1-c1e2-0-0
Degree $4$
Conductor $2483776$
Sign $1$
Analytic cond. $158.367$
Root an. cond. $3.54745$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 6·9-s + 8·11-s − 4·16-s − 16·17-s + 12·18-s − 6·19-s − 16·22-s − 10·25-s + 8·32-s + 32·34-s − 12·36-s + 12·38-s + 18·41-s + 2·43-s + 16·44-s − 5·49-s + 20·50-s − 8·64-s − 20·67-s − 32·68-s + 12·73-s − 12·76-s + 27·81-s − 36·82-s − 14·83-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 2·9-s + 2.41·11-s − 16-s − 3.88·17-s + 2.82·18-s − 1.37·19-s − 3.41·22-s − 2·25-s + 1.41·32-s + 5.48·34-s − 2·36-s + 1.94·38-s + 2.81·41-s + 0.304·43-s + 2.41·44-s − 5/7·49-s + 2.82·50-s − 64-s − 2.44·67-s − 3.88·68-s + 1.40·73-s − 1.37·76-s + 3·81-s − 3.97·82-s − 1.53·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2483776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2483776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2483776\)    =    \(2^{6} \cdot 197^{2}\)
Sign: $1$
Analytic conductor: \(158.367\)
Root analytic conductor: \(3.54745\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2483776,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
197$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−7.14302530222668133337765646449, −6.92472041340395206972505718477, −6.29130249337217335424693461714, −6.17591940844279569664203110179, −6.01799931951343104662762034245, −5.05536963507269575186847712732, −4.32966566168173965152585313793, −4.15785519193072071590213469433, −3.93436739100015316068436710765, −2.80499269979292397198625812082, −2.33277116067346011831864956690, −2.04602760884368616612640052482, −1.29204853850579349824121166688, 0, 0, 1.29204853850579349824121166688, 2.04602760884368616612640052482, 2.33277116067346011831864956690, 2.80499269979292397198625812082, 3.93436739100015316068436710765, 4.15785519193072071590213469433, 4.32966566168173965152585313793, 5.05536963507269575186847712732, 6.01799931951343104662762034245, 6.17591940844279569664203110179, 6.29130249337217335424693461714, 6.92472041340395206972505718477, 7.14302530222668133337765646449

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