Properties

Label 4-2e16-1.1-c2e2-0-2
Degree $4$
Conductor $65536$
Sign $1$
Analytic cond. $48.6575$
Root an. cond. $2.64111$
Motivic weight $2$
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  − 14·9-s − 4·17-s − 50·25-s + 92·41-s + 98·49-s − 284·73-s + 115·81-s + 292·89-s + 188·97-s + 196·113-s − 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 56·153-s + 157-s + 163-s + 167-s − 338·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 1.55·9-s − 0.235·17-s − 2·25-s + 2.24·41-s + 2·49-s − 3.89·73-s + 1.41·81-s + 3.28·89-s + 1.93·97-s + 1.73·113-s − 0.380·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.366·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65536\)    =    \(2^{16}\)
Sign: $1$
Analytic conductor: \(48.6575\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65536,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.263102930\)
\(L(\frac12)\) \(\approx\) \(1.263102930\)
\(L(2)\) 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$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \)
5$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
11$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 46 T + p^{2} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
59$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \)
61$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 62 T + p^{2} T^{2} )( 1 + 62 T + p^{2} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 142 T + p^{2} T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2$ \( ( 1 - 158 T + p^{2} T^{2} )( 1 + 158 T + p^{2} T^{2} ) \)
89$C_2$ \( ( 1 - 146 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 94 T + p^{2} T^{2} )^{2} \)
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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

−11.95832235342697138385429777608, −11.61821681805902930135571565009, −11.12924304862882124931113419963, −10.66973369449957511001115338546, −10.17605017827648353781596921762, −9.591098443176455485018716721476, −8.994638140578282569390533968581, −8.791790721916551867271629420340, −8.167377764487528629575313834601, −7.44085226979205323381891257270, −7.41620216675404220918043787182, −6.24817934806743903808624411262, −5.90340836566319811569626513063, −5.65858282120675127811390691224, −4.73914625241094551628794662237, −4.12429886404205156399970347960, −3.41827492635768835011073277830, −2.65318984815480818688382546493, −2.01985653923667492585051242514, −0.56817186997133595885226430646, 0.56817186997133595885226430646, 2.01985653923667492585051242514, 2.65318984815480818688382546493, 3.41827492635768835011073277830, 4.12429886404205156399970347960, 4.73914625241094551628794662237, 5.65858282120675127811390691224, 5.90340836566319811569626513063, 6.24817934806743903808624411262, 7.41620216675404220918043787182, 7.44085226979205323381891257270, 8.167377764487528629575313834601, 8.791790721916551867271629420340, 8.994638140578282569390533968581, 9.591098443176455485018716721476, 10.17605017827648353781596921762, 10.66973369449957511001115338546, 11.12924304862882124931113419963, 11.61821681805902930135571565009, 11.95832235342697138385429777608

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