Properties

Label 4-3e4-1.1-c17e2-0-1
Degree $4$
Conductor $81$
Sign $1$
Analytic cond. $271.919$
Root an. cond. $4.06078$
Motivic weight $17$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.31e5·4-s − 3.92e5·7-s − 2.61e9·13-s + 8.38e6·16-s − 2.22e11·19-s − 1.51e12·25-s + 5.15e10·28-s + 8.14e12·31-s + 3.43e13·37-s − 1.20e14·43-s − 4.65e14·49-s + 3.43e14·52-s + 1.40e15·61-s + 2.25e15·64-s − 4.36e15·67-s − 2.18e16·73-s + 2.91e16·76-s − 3.21e16·79-s + 1.02e15·91-s − 1.09e17·97-s + 1.98e17·100-s − 2.51e17·103-s − 3.10e17·109-s − 3.29e12·112-s + 2.14e17·121-s − 1.06e18·124-s + ⋯
L(s)  = 1  − 1.00·4-s − 0.0257·7-s − 0.889·13-s + 0.000488·16-s − 3.00·19-s − 1.98·25-s + 0.0257·28-s + 1.71·31-s + 1.60·37-s − 1.57·43-s − 1.99·49-s + 0.890·52-s + 0.938·61-s + 0.999·64-s − 1.31·67-s − 3.17·73-s + 3.00·76-s − 2.38·79-s + 0.0229·91-s − 1.41·97-s + 1.98·100-s − 1.95·103-s − 1.49·109-s − 1.25e − 5·112-s + 0.423·121-s − 1.71·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(271.919\)
Root analytic conductor: \(4.06078\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 81,\ (\ :17/2, 17/2),\ 1)\)

Particular Values

\(L(9)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{19}{2})\) 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)$
bad3 \( 1 \)
good2$C_2^2$ \( 1 + 4097 p^{5} T^{2} + p^{34} T^{4} \)
5$C_2^2$ \( 1 + 12124842962 p^{3} T^{2} + p^{34} T^{4} \)
7$C_2$ \( ( 1 + 28060 p T + p^{17} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 214059915897156458 T^{2} + p^{34} T^{4} \)
13$C_2$ \( ( 1 + 100663990 p T + p^{17} T^{2} )^{2} \)
17$C_2^2$ \( 1 + \)\(43\!\cdots\!94\)\( T^{2} + p^{34} T^{4} \)
19$C_2$ \( ( 1 + 111322541176 T + p^{17} T^{2} )^{2} \)
23$C_2^2$ \( 1 + \)\(16\!\cdots\!66\)\( T^{2} + p^{34} T^{4} \)
29$C_2^2$ \( 1 + \)\(30\!\cdots\!18\)\( T^{2} + p^{34} T^{4} \)
31$C_2$ \( ( 1 - 4073718654908 T + p^{17} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 17155445649590 T + p^{17} T^{2} )^{2} \)
41$C_2^2$ \( 1 + \)\(28\!\cdots\!62\)\( T^{2} + p^{34} T^{4} \)
43$C_2$ \( ( 1 + 60341472064120 T + p^{17} T^{2} )^{2} \)
47$C_2^2$ \( 1 + \)\(15\!\cdots\!34\)\( T^{2} + p^{34} T^{4} \)
53$C_2^2$ \( 1 + \)\(31\!\cdots\!66\)\( T^{2} + p^{34} T^{4} \)
59$C_2^2$ \( 1 + \)\(24\!\cdots\!38\)\( T^{2} + p^{34} T^{4} \)
61$C_2$ \( ( 1 - 702559953764942 T + p^{17} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2182028305039120 T + p^{17} T^{2} )^{2} \)
71$C_2^2$ \( 1 - \)\(31\!\cdots\!18\)\( T^{2} + p^{34} T^{4} \)
73$C_2$ \( ( 1 + 10947509924726410 T + p^{17} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16061998246908484 T + p^{17} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(10\!\cdots\!14\)\( T^{2} + p^{34} T^{4} \)
89$C_2^2$ \( 1 + \)\(15\!\cdots\!58\)\( T^{2} + p^{34} T^{4} \)
97$C_2$ \( ( 1 + 54656424308957170 T + p^{17} T^{2} )^{2} \)
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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

−16.81622667598507765512526477717, −15.85947537659305065213433989527, −14.87526446089042273355320184633, −14.58449543458420276848242202145, −13.25629362199486821778746100212, −13.25561137106467077185716243682, −12.12477482030216004370621508205, −11.30052842808206774475010311875, −10.14250724063826086247272492793, −9.673701821716499426555161868049, −8.556721042763347732962795351257, −8.077348028145380647221480505878, −6.73134216904321289019257692850, −5.87324591486602368867981996961, −4.45044005683847321545305174941, −4.31574438729222029538685700996, −2.71633511119956943211308443294, −1.71418291356430341740188768655, 0, 0, 1.71418291356430341740188768655, 2.71633511119956943211308443294, 4.31574438729222029538685700996, 4.45044005683847321545305174941, 5.87324591486602368867981996961, 6.73134216904321289019257692850, 8.077348028145380647221480505878, 8.556721042763347732962795351257, 9.673701821716499426555161868049, 10.14250724063826086247272492793, 11.30052842808206774475010311875, 12.12477482030216004370621508205, 13.25561137106467077185716243682, 13.25629362199486821778746100212, 14.58449543458420276848242202145, 14.87526446089042273355320184633, 15.85947537659305065213433989527, 16.81622667598507765512526477717

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