Properties

Label 4-2e16-1.1-c1e2-0-6
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  + 8·7-s + 2·9-s − 4·17-s − 8·23-s + 6·25-s + 12·41-s − 16·47-s + 34·49-s + 16·63-s − 24·71-s − 28·73-s − 16·79-s − 5·81-s + 4·89-s − 4·97-s + 8·103-s + 4·113-s − 32·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + ⋯
L(s)  = 1  + 3.02·7-s + 2/3·9-s − 0.970·17-s − 1.66·23-s + 6/5·25-s + 1.87·41-s − 2.33·47-s + 34/7·49-s + 2.01·63-s − 2.84·71-s − 3.27·73-s − 1.80·79-s − 5/9·81-s + 0.423·89-s − 0.406·97-s + 0.788·103-s + 0.376·113-s − 2.93·119-s + 1.63·121-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 − 0.646·153-s + 0.0798·157-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: induced by $\chi_{256} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.036627835\)
\(L(\frac12)\) \(\approx\) \(2.036627835\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−11.95229495174916424715383421785, −11.75762837471476313294134706289, −11.15648399208933492725911632942, −11.14494699170078649725293233429, −10.26577727041265197797122910912, −10.19846153151799484232110380476, −9.256108972438302022834978407199, −8.611783944663925494202640188699, −8.468954768123919982906953682469, −7.86528301643183403367527227627, −7.42675807818385756312028625007, −7.03622164116349230603131877481, −6.05194408029281576944524179606, −5.65174324111818920691717656115, −4.67760979703143973951450660168, −4.59266733615629306970623763094, −4.18750880843649192496613275121, −2.86673688421774418436559376471, −1.84807485602538521290021290184, −1.48527687479949932962438323565, 1.48527687479949932962438323565, 1.84807485602538521290021290184, 2.86673688421774418436559376471, 4.18750880843649192496613275121, 4.59266733615629306970623763094, 4.67760979703143973951450660168, 5.65174324111818920691717656115, 6.05194408029281576944524179606, 7.03622164116349230603131877481, 7.42675807818385756312028625007, 7.86528301643183403367527227627, 8.468954768123919982906953682469, 8.611783944663925494202640188699, 9.256108972438302022834978407199, 10.19846153151799484232110380476, 10.26577727041265197797122910912, 11.14494699170078649725293233429, 11.15648399208933492725911632942, 11.75762837471476313294134706289, 11.95229495174916424715383421785

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