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Properties

Label 4-788e2-1.1-c1e2-0-0
Degree $4$
Conductor $620944$
Sign $-1$
Analytic cond. $39.5919$
Root an. cond. $2.50842$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 6·9-s − 4·13-s − 4·16-s − 16·17-s + 12·18-s − 10·25-s + 8·26-s + 14·29-s + 8·32-s + 32·34-s − 12·36-s + 14·37-s + 18·41-s − 5·49-s + 20·50-s − 8·52-s + 20·53-s − 28·58-s + 10·61-s − 8·64-s − 32·68-s + 12·73-s − 28·74-s + 27·81-s − 36·82-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 2·9-s − 1.10·13-s − 16-s − 3.88·17-s + 2.82·18-s − 2·25-s + 1.56·26-s + 2.59·29-s + 1.41·32-s + 5.48·34-s − 2·36-s + 2.30·37-s + 2.81·41-s − 5/7·49-s + 2.82·50-s − 1.10·52-s + 2.74·53-s − 3.67·58-s + 1.28·61-s − 64-s − 3.88·68-s + 1.40·73-s − 3.25·74-s + 3·81-s − 3.97·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 620944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 620944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(620944\)    =    \(2^{4} \cdot 197^{2}\)
Sign: $-1$
Analytic conductor: \(39.5919\)
Root analytic conductor: \(2.50842\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 620944,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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

−8.282753935470816342481418791481, −8.056603231480438386460344076725, −7.33956567594263867864342447503, −6.92472041340395206972505718477, −6.37618381971660332853552601773, −6.17591940844279569664203110179, −5.51910959519898081306299022355, −4.84286071876505944866542752996, −4.21913762037456798202222334655, −4.15785519193072071590213469433, −2.60609363958670334343605384522, −2.44365137125241217402163994690, −2.33277116067346011831864956690, −0.74177520741704320779285872897, 0, 0.74177520741704320779285872897, 2.33277116067346011831864956690, 2.44365137125241217402163994690, 2.60609363958670334343605384522, 4.15785519193072071590213469433, 4.21913762037456798202222334655, 4.84286071876505944866542752996, 5.51910959519898081306299022355, 6.17591940844279569664203110179, 6.37618381971660332853552601773, 6.92472041340395206972505718477, 7.33956567594263867864342447503, 8.056603231480438386460344076725, 8.282753935470816342481418791481

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