Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s − 4·13-s − 4·16-s + 9·25-s − 5·27-s − 4·39-s − 12·43-s − 4·48-s − 10·49-s − 24·61-s + 9·75-s + 20·79-s + 81-s − 32·103-s + 8·117-s − 121-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s − 1.10·13-s − 16-s + 9/5·25-s − 0.962·27-s − 0.640·39-s − 1.82·43-s − 0.577·48-s − 1.42·49-s − 3.07·61-s + 1.03·75-s + 2.25·79-s + 1/9·81-s − 3.15·103-s + 0.739·117-s − 0.0909·121-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: \(1\)
Selberg data: \((4,\ 184041,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
11$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.5.a_aj
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.a_n
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.a_cn
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.43.m_es
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 93 T^{2} + p^{2} T^{4} \) 2.59.a_adp
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.61.y_kg
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.a_dh
71$C_2^2$ \( 1 - 133 T^{2} + p^{2} T^{4} \) 2.71.a_afd
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.a_fa
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.89.a_bv
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.a_fp
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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

−8.948120652585820244130975284360, −8.461555920787699533888934782741, −7.964317948425853675493648350826, −7.62304324726744720688675232602, −6.80871386872109234847475322846, −6.70397308989938655889499709405, −6.02893867211806778724555779842, −5.20490759227891415189036285030, −4.90535755554277169598852397879, −4.40554964744680566134785437844, −3.47730634993023562727955973850, −2.96900256953584074650273461463, −2.45956532354583680984131227863, −1.60583225677726134700355751249, 0, 1.60583225677726134700355751249, 2.45956532354583680984131227863, 2.96900256953584074650273461463, 3.47730634993023562727955973850, 4.40554964744680566134785437844, 4.90535755554277169598852397879, 5.20490759227891415189036285030, 6.02893867211806778724555779842, 6.70397308989938655889499709405, 6.80871386872109234847475322846, 7.62304324726744720688675232602, 7.964317948425853675493648350826, 8.461555920787699533888934782741, 8.948120652585820244130975284360

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