Properties

Label 4-429e2-1.1-c1e2-0-1
Degree $4$
Conductor $184041$
Sign $1$
Analytic cond. $11.7346$
Root an. cond. $1.85083$
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  + 4-s + 9-s − 2·13-s − 3·16-s + 2·25-s + 8·29-s + 36-s + 4·43-s + 10·49-s − 2·52-s + 12·53-s + 4·61-s − 7·64-s + 4·79-s + 81-s + 2·100-s − 24·101-s + 12·113-s + 8·116-s − 2·117-s + 121-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s − 0.554·13-s − 3/4·16-s + 2/5·25-s + 1.48·29-s + 1/6·36-s + 0.609·43-s + 10/7·49-s − 0.277·52-s + 1.64·53-s + 0.512·61-s − 7/8·64-s + 0.450·79-s + 1/9·81-s + 1/5·100-s − 2.38·101-s + 1.12·113-s + 0.742·116-s − 0.184·117-s + 1/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(184041\)    =    \(3^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(11.7346\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 184041,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.854640035\)
\(L(\frac12)\) \(\approx\) \(1.854640035\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.19.a_as
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ai_bm
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.37.a_cc
41$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.41.a_o
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.47.a_as
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.ae_ck
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.71.a_ck
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.ae_ck
83$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.83.a_cc
89$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \) 2.89.a_afi
97$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.97.a_o
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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

−9.034675924042442143095749364854, −8.751255841935656095454046788480, −8.201720666313246234236012947159, −7.64207526479637796430779583830, −7.06106563280908413208522152193, −6.89083570242035191053138253645, −6.32264419475329643065103807654, −5.66441761296988050389009552569, −5.19940310794129448841020654032, −4.43271259079955941847260762295, −4.19246927836238336661479687820, −3.23556710348062248878122127648, −2.58345108108264936945916302095, −2.06329049460713332419606765221, −0.899003034750625609668785182244, 0.899003034750625609668785182244, 2.06329049460713332419606765221, 2.58345108108264936945916302095, 3.23556710348062248878122127648, 4.19246927836238336661479687820, 4.43271259079955941847260762295, 5.19940310794129448841020654032, 5.66441761296988050389009552569, 6.32264419475329643065103807654, 6.89083570242035191053138253645, 7.06106563280908413208522152193, 7.64207526479637796430779583830, 8.201720666313246234236012947159, 8.751255841935656095454046788480, 9.034675924042442143095749364854

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