Properties

Label 4-105e2-1.1-c1e2-0-8
Degree $4$
Conductor $11025$
Sign $1$
Analytic cond. $0.702963$
Root an. cond. $0.915657$
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  + 3·4-s + 2·5-s − 9-s − 12·11-s + 5·16-s + 12·19-s + 6·20-s − 25-s + 4·29-s − 20·31-s − 3·36-s + 4·41-s − 36·44-s − 2·45-s − 49-s − 24·55-s + 16·59-s − 4·61-s + 3·64-s + 20·71-s + 36·76-s − 8·79-s + 10·80-s + 81-s − 12·89-s + 24·95-s + 12·99-s + ⋯
L(s)  = 1  + 3/2·4-s + 0.894·5-s − 1/3·9-s − 3.61·11-s + 5/4·16-s + 2.75·19-s + 1.34·20-s − 1/5·25-s + 0.742·29-s − 3.59·31-s − 1/2·36-s + 0.624·41-s − 5.42·44-s − 0.298·45-s − 1/7·49-s − 3.23·55-s + 2.08·59-s − 0.512·61-s + 3/8·64-s + 2.37·71-s + 4.12·76-s − 0.900·79-s + 1.11·80-s + 1/9·81-s − 1.27·89-s + 2.46·95-s + 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.702963\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{105} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.348340487\)
\(L(\frac12)\) \(\approx\) \(1.348340487\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−14.07765741036902913605419866632, −13.34777610896378407247861108327, −13.03878561228530477495552939210, −12.59576519015341727507158702442, −11.86948020238545760806007327417, −11.27433118061208772548131650751, −10.69192697832386041230196696032, −10.65843719583477371813672800041, −9.704083858777687059690286950373, −9.590614160303390257499736002489, −8.370119229283450604227174471651, −7.77449242194542902577294645997, −7.40285334522283855310331904915, −6.96295836787847011215570227142, −5.66673716415926716727854405042, −5.53092384774540902824306223888, −5.20934549967759721253207672094, −3.31926418578659151487141795523, −2.70173620439600678454391775595, −2.06135419027838012408223731157, 2.06135419027838012408223731157, 2.70173620439600678454391775595, 3.31926418578659151487141795523, 5.20934549967759721253207672094, 5.53092384774540902824306223888, 5.66673716415926716727854405042, 6.96295836787847011215570227142, 7.40285334522283855310331904915, 7.77449242194542902577294645997, 8.370119229283450604227174471651, 9.590614160303390257499736002489, 9.704083858777687059690286950373, 10.65843719583477371813672800041, 10.69192697832386041230196696032, 11.27433118061208772548131650751, 11.86948020238545760806007327417, 12.59576519015341727507158702442, 13.03878561228530477495552939210, 13.34777610896378407247861108327, 14.07765741036902913605419866632

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