Properties

Label 4-2e16-1.1-c1e2-0-2
Degree $4$
Conductor $65536$
Sign $1$
Analytic cond. $4.17863$
Root an. cond. $1.42974$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65536\)    =    \(2^{16}\)
Sign: $1$
Analytic conductor: \(4.17863\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586248366\)
\(L(\frac12)\) \(\approx\) \(1.586248366\)
\(L(\frac{3}{2})\) 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

−12.35797581564216331802133964919, −11.75655140830289064450817435587, −11.43416050332192748416656544847, −10.80930439699301989240452536246, −10.13890570115710586872097299926, −9.938036229146395142845664532292, −9.536453957000876163766901881878, −9.033934658635621728990784548147, −8.066335838706079171976710141123, −7.85204783264984617615072264980, −7.55299395754220445658708645745, −6.84371134663949782422454400794, −6.00140065324702785753059105988, −5.80484397546847799453205431959, −5.09546563559245222823656645910, −4.39774330912317846963186570778, −3.64183884460257760595955162875, −3.22834670546030511963807300880, −2.09259386172801032688627778191, −1.13265678082836875231989401788, 1.13265678082836875231989401788, 2.09259386172801032688627778191, 3.22834670546030511963807300880, 3.64183884460257760595955162875, 4.39774330912317846963186570778, 5.09546563559245222823656645910, 5.80484397546847799453205431959, 6.00140065324702785753059105988, 6.84371134663949782422454400794, 7.55299395754220445658708645745, 7.85204783264984617615072264980, 8.066335838706079171976710141123, 9.033934658635621728990784548147, 9.536453957000876163766901881878, 9.938036229146395142845664532292, 10.13890570115710586872097299926, 10.80930439699301989240452536246, 11.43416050332192748416656544847, 11.75655140830289064450817435587, 12.35797581564216331802133964919

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