Properties

Label 4-1152e2-1.1-c1e2-0-27
Degree $4$
Conductor $1327104$
Sign $1$
Analytic cond. $84.6173$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·13-s + 10·25-s − 4·37-s − 10·49-s − 4·61-s + 20·73-s + 12·97-s − 4·109-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3.32·13-s + 2·25-s − 0.657·37-s − 1.42·49-s − 0.512·61-s + 2.34·73-s + 1.21·97-s − 0.383·109-s − 0.545·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1327104\)    =    \(2^{14} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(84.6173\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1327104,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.861725379\)
\(L(\frac12)\) \(\approx\) \(2.861725379\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 174 T^{2} + p^{2} T^{4} \)
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

−8.137823048601241287533732648958, −7.66120262089958281679783825752, −6.86534925988280107121181602054, −6.63956777233333377219867681148, −6.33602888777443961894949754608, −5.83321691623881889608390550552, −5.38230484758393479905549018953, −4.85744268609471259420249269277, −4.32082038759074342163981002570, −3.73973402676937641544434381414, −3.35681809369203953352020402641, −3.01010144569182544770623760102, −2.02762039368627548909510662259, −1.35876606762133764401638958608, −0.851048518166312469994206051827, 0.851048518166312469994206051827, 1.35876606762133764401638958608, 2.02762039368627548909510662259, 3.01010144569182544770623760102, 3.35681809369203953352020402641, 3.73973402676937641544434381414, 4.32082038759074342163981002570, 4.85744268609471259420249269277, 5.38230484758393479905549018953, 5.83321691623881889608390550552, 6.33602888777443961894949754608, 6.63956777233333377219867681148, 6.86534925988280107121181602054, 7.66120262089958281679783825752, 8.137823048601241287533732648958

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