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Properties

Label 4-114075-1.1-c1e2-0-19
Degree $4$
Conductor $114075$
Sign $-1$
Analytic cond. $7.27352$
Root an. cond. $1.64223$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s + 9-s − 12-s − 4·13-s − 3·16-s − 6·23-s + 25-s + 27-s − 12·29-s − 36-s − 4·39-s + 10·43-s − 3·48-s − 10·49-s + 4·52-s − 12·53-s − 8·61-s + 7·64-s − 6·69-s + 75-s + 4·79-s + 81-s − 12·87-s + 6·92-s − 100-s + 22·103-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 3/4·16-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 2.22·29-s − 1/6·36-s − 0.640·39-s + 1.52·43-s − 0.433·48-s − 1.42·49-s + 0.554·52-s − 1.64·53-s − 1.02·61-s + 7/8·64-s − 0.722·69-s + 0.115·75-s + 0.450·79-s + 1/9·81-s − 1.28·87-s + 0.625·92-s − 0.0999·100-s + 2.16·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(114075\)    =    \(3^{3} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(7.27352\)
Root analytic conductor: \(1.64223\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 114075,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.g_bu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.37.a_bu
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.43.ak_dy
47$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.47.a_abm
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.71.a_ao
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.73.a_ac
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ae_ew
83$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.83.a_bi
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.89.a_de
97$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.97.a_bi
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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

−9.243883739655684232285086452642, −8.971258763318936071234614278231, −8.116802337879264713575699579331, −7.83442737994774170091406163375, −7.40703833223320554641736294118, −6.82606790356779222452957979984, −6.19209914820645800528883004008, −5.62735221828977959334464775459, −4.93773160626721556914727239569, −4.47284619946525654223380577953, −3.90261330422601089705278577698, −3.22736973636347109410824917240, −2.38001630660605951551548226436, −1.76747923206471972685248272428, 0, 1.76747923206471972685248272428, 2.38001630660605951551548226436, 3.22736973636347109410824917240, 3.90261330422601089705278577698, 4.47284619946525654223380577953, 4.93773160626721556914727239569, 5.62735221828977959334464775459, 6.19209914820645800528883004008, 6.82606790356779222452957979984, 7.40703833223320554641736294118, 7.83442737994774170091406163375, 8.116802337879264713575699579331, 8.971258763318936071234614278231, 9.243883739655684232285086452642

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