↑

Properties

Label 4-1155e2-1.1-c1e2-0-5
Degree $4$
Conductor $1334025$
Sign $1$
Analytic cond. $85.0585$
Root an. cond. $3.03689$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 4·5-s − 9-s + 2·11-s + 5·16-s − 12·20-s + 11·25-s − 12·31-s − 3·36-s − 12·41-s + 6·44-s + 4·45-s − 49-s − 8·55-s + 4·61-s + 3·64-s + 16·71-s − 4·79-s − 20·80-s + 81-s + 24·89-s − 2·99-s + 33·100-s − 28·101-s + 36·109-s + 3·121-s − 36·124-s + ⋯
L(s)  = 1  + 3/2·4-s − 1.78·5-s − 1/3·9-s + 0.603·11-s + 5/4·16-s − 2.68·20-s + 11/5·25-s − 2.15·31-s − 1/2·36-s − 1.87·41-s + 0.904·44-s + 0.596·45-s − 1/7·49-s − 1.07·55-s + 0.512·61-s + 3/8·64-s + 1.89·71-s − 0.450·79-s − 2.23·80-s + 1/9·81-s + 2.54·89-s − 0.201·99-s + 3.29·100-s − 2.78·101-s + 3.44·109-s + 3/11·121-s − 3.23·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1334025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(85.0585\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1334025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.689591229\)
\(L(\frac12)\) \(\approx\) \(1.689591229\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.79.e_gg
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \) 2.97.a_agc
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.19919550171989171630485507957, −9.535734241313627084831169393116, −9.110530735907319921264441673417, −8.626337533542103104125760790691, −8.344588168152005753671927811958, −7.69727475863075152313665627912, −7.65130469186450579823583422549, −6.97251080753238389549594678385, −6.91541796721221392793059397525, −6.38533601026859193980550670572, −5.93670099945082156465073246983, −5.11299480821682757260827718333, −5.07551724314611903166488882426, −4.06802003649711077096787991795, −3.82994388585028736148059895500, −3.30484578069762892950136960942, −2.94648901688628314398379962575, −2.10154163342333559636944188902, −1.61859760466338375287059253568, −0.55170394821371558821969932425, 0.55170394821371558821969932425, 1.61859760466338375287059253568, 2.10154163342333559636944188902, 2.94648901688628314398379962575, 3.30484578069762892950136960942, 3.82994388585028736148059895500, 4.06802003649711077096787991795, 5.07551724314611903166488882426, 5.11299480821682757260827718333, 5.93670099945082156465073246983, 6.38533601026859193980550670572, 6.91541796721221392793059397525, 6.97251080753238389549594678385, 7.65130469186450579823583422549, 7.69727475863075152313665627912, 8.344588168152005753671927811958, 8.626337533542103104125760790691, 9.110530735907319921264441673417, 9.535734241313627084831169393116, 10.19919550171989171630485507957

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