Properties

Label 4-1792e2-1.1-c1e2-0-1
Degree $4$
Conductor $3211264$
Sign $1$
Analytic cond. $204.752$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 6·9-s − 12·17-s + 6·25-s − 16·31-s − 4·41-s + 16·47-s + 3·49-s − 12·63-s − 16·71-s − 20·73-s − 32·79-s + 27·81-s + 12·89-s − 12·97-s − 32·103-s + 4·113-s + 24·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 72·153-s + 157-s + ⋯
L(s)  = 1  − 0.755·7-s + 2·9-s − 2.91·17-s + 6/5·25-s − 2.87·31-s − 0.624·41-s + 2.33·47-s + 3/7·49-s − 1.51·63-s − 1.89·71-s − 2.34·73-s − 3.60·79-s + 3·81-s + 1.27·89-s − 1.21·97-s − 3.15·103-s + 0.376·113-s + 2.20·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 5.82·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3211264\)    =    \(2^{16} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(204.752\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1792} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3211264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.318146764\)
\(L(\frac12)\) \(\approx\) \(1.318146764\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.745836656439833316282704051939, −8.903712408776508411217315203105, −8.890938324018158262513287189505, −8.582266227698537095806042349993, −7.70571146092749935795184798176, −7.25214299997128106071549019140, −7.11196304633871858201230720088, −6.84466491763944435938810317878, −6.50588879532471922078838354857, −5.68784206126345907746742174977, −5.67284219814268850127895572559, −4.78320428605617232983467798508, −4.42345508652110506433275110732, −4.13262069038847401481089269234, −3.83905909196373736255552550968, −2.98388876822502447763400489625, −2.61355728361102460213543875210, −1.69115921376259029597674795708, −1.67703211091000580964232125707, −0.41698006543836759305244622919, 0.41698006543836759305244622919, 1.67703211091000580964232125707, 1.69115921376259029597674795708, 2.61355728361102460213543875210, 2.98388876822502447763400489625, 3.83905909196373736255552550968, 4.13262069038847401481089269234, 4.42345508652110506433275110732, 4.78320428605617232983467798508, 5.67284219814268850127895572559, 5.68784206126345907746742174977, 6.50588879532471922078838354857, 6.84466491763944435938810317878, 7.11196304633871858201230720088, 7.25214299997128106071549019140, 7.70571146092749935795184798176, 8.582266227698537095806042349993, 8.890938324018158262513287189505, 8.903712408776508411217315203105, 9.745836656439833316282704051939

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