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Properties

Label 4-8712e2-1.1-c1e2-0-3
Degree $4$
Conductor $75898944$
Sign $1$
Analytic cond. $4839.38$
Root an. cond. $8.34060$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 2·7-s + 2·13-s + 4·17-s + 8·19-s − 9·23-s + 25-s − 2·29-s − 7·31-s − 6·35-s − 11·37-s + 6·41-s + 6·43-s − 16·47-s + 6·49-s − 8·53-s + 5·59-s + 6·61-s − 6·65-s + 15·67-s + 5·71-s − 2·73-s + 14·79-s + 10·83-s − 12·85-s + 7·89-s + 4·91-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.755·7-s + 0.554·13-s + 0.970·17-s + 1.83·19-s − 1.87·23-s + 1/5·25-s − 0.371·29-s − 1.25·31-s − 1.01·35-s − 1.80·37-s + 0.937·41-s + 0.914·43-s − 2.33·47-s + 6/7·49-s − 1.09·53-s + 0.650·59-s + 0.768·61-s − 0.744·65-s + 1.83·67-s + 0.593·71-s − 0.234·73-s + 1.57·79-s + 1.09·83-s − 1.30·85-s + 0.741·89-s + 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75898944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75898944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(75898944\)    =    \(2^{6} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4839.38\)
Root analytic conductor: \(8.34060\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 75898944,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.252098973\)
\(L(\frac12)\) \(\approx\) \(2.252098973\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ac
13$C_4$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_k
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$D_{4}$ \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.23.j_ck
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_bq
31$D_{4}$ \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.31.h_cs
37$D_{4}$ \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.37.l_dw
41$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_cw
43$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_da
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_cc
59$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.59.af_s
61$D_{4}$ \( 1 - 6 T + 114 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_ek
67$D_{4}$ \( 1 - 15 T + 186 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.67.ap_he
71$D_{4}$ \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.71.af_eg
73$D_{4}$ \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fa
79$D_{4}$ \( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.79.ao_hi
83$D_{4}$ \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_gs
89$D_{4}$ \( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.89.ah_fw
97$D_{4}$ \( 1 - 27 T + 372 T^{2} - 27 p T^{3} + p^{2} T^{4} \) 2.97.abb_oi
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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

−7.83759616300998346646168961788, −7.72036104928894808309883722389, −7.38468642085246602758979611203, −7.12442818796937239745216254711, −6.48405108424065690282867098688, −6.33467667536438441401040690894, −5.77503212644068543761214247225, −5.44034047253880150875150526335, −5.13260006245767723164161409460, −4.95346993736919533949863942038, −4.34087531986733407334779298907, −3.86629603531077187530636678049, −3.58910511603259224570815072356, −3.57948479042599992143172610233, −3.06936253573987295863097995127, −2.33754167438792186562776983421, −1.87019642571345915445113381774, −1.59139422094954619345736616727, −0.841511632848326255381804365901, −0.44146815386578294920970869460, 0.44146815386578294920970869460, 0.841511632848326255381804365901, 1.59139422094954619345736616727, 1.87019642571345915445113381774, 2.33754167438792186562776983421, 3.06936253573987295863097995127, 3.57948479042599992143172610233, 3.58910511603259224570815072356, 3.86629603531077187530636678049, 4.34087531986733407334779298907, 4.95346993736919533949863942038, 5.13260006245767723164161409460, 5.44034047253880150875150526335, 5.77503212644068543761214247225, 6.33467667536438441401040690894, 6.48405108424065690282867098688, 7.12442818796937239745216254711, 7.38468642085246602758979611203, 7.72036104928894808309883722389, 7.83759616300998346646168961788

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