Properties

Label 4-2e12-1.1-c1e2-0-1
Degree $4$
Conductor $4096$
Sign $1$
Analytic cond. $0.261164$
Root an. cond. $0.714872$
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  + 2·9-s − 12·17-s + 10·25-s − 12·41-s − 14·49-s + 4·73-s − 5·81-s + 36·89-s + 20·97-s + 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s − 2.91·17-s + 2·25-s − 1.87·41-s − 2·49-s + 0.468·73-s − 5/9·81-s + 3.81·89-s + 2.03·97-s + 3.38·113-s − 1.27·121-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4096\)    =    \(2^{12}\)
Sign: $1$
Analytic conductor: \(0.261164\)
Root analytic conductor: \(0.714872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{64} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4096,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7931241833\)
\(L(\frac12)\) \(\approx\) \(0.7931241833\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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

−15.24061031761495894594432737469, −14.77666960937088281402227755553, −14.11744779141590763691021382241, −13.40416631353195959623048426493, −13.01138251672671175392876917229, −12.71928374289623567725367707515, −11.74893177195257215025275776857, −11.32208652688223242730440004648, −10.65483692056500405743349122518, −10.26762558111546507012807907342, −9.334876857942576191292204419070, −8.852476829446497820961864119358, −8.351563371273259363456532004580, −7.34996069932697346126672149293, −6.66435165558541062030334206319, −6.37961207777060964519609837006, −4.85982387481815205334569253252, −4.66052578287437967441098070770, −3.40294200831321822698668003934, −2.08774265100019936984633203220, 2.08774265100019936984633203220, 3.40294200831321822698668003934, 4.66052578287437967441098070770, 4.85982387481815205334569253252, 6.37961207777060964519609837006, 6.66435165558541062030334206319, 7.34996069932697346126672149293, 8.351563371273259363456532004580, 8.852476829446497820961864119358, 9.334876857942576191292204419070, 10.26762558111546507012807907342, 10.65483692056500405743349122518, 11.32208652688223242730440004648, 11.74893177195257215025275776857, 12.71928374289623567725367707515, 13.01138251672671175392876917229, 13.40416631353195959623048426493, 14.11744779141590763691021382241, 14.77666960937088281402227755553, 15.24061031761495894594432737469

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