Properties

Label 4-33e4-1.1-c5e2-0-2
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $30505.4$
Root an. cond. $13.2158$
Motivic weight $5$
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  − 44·4-s + 196·5-s + 912·16-s − 8.62e3·20-s − 752·23-s + 2.25e4·25-s + 7.44e3·31-s − 1.89e4·37-s − 1.03e4·47-s + 2.25e4·49-s + 6.89e4·53-s + 3.28e4·59-s + 4.92e3·64-s + 4.22e4·67-s + 1.09e5·71-s + 1.78e5·80-s − 1.13e4·89-s + 3.30e4·92-s − 5.18e4·97-s − 9.92e5·100-s − 3.04e5·103-s − 3.31e5·113-s − 1.47e5·115-s − 3.27e5·124-s + 1.92e6·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.37·4-s + 3.50·5-s + 0.890·16-s − 4.82·20-s − 0.296·23-s + 7.21·25-s + 1.39·31-s − 2.27·37-s − 0.685·47-s + 1.34·49-s + 3.37·53-s + 1.22·59-s + 0.150·64-s + 1.14·67-s + 2.58·71-s + 3.12·80-s − 0.151·89-s + 0.407·92-s − 0.559·97-s − 9.92·100-s − 2.82·103-s − 2.44·113-s − 1.03·115-s − 1.91·124-s + 11.0·125-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(30505.4\)
Root analytic conductor: \(13.2158\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(6.336897483\)
\(L(\frac12)\) \(\approx\) \(6.336897483\)
\(L(\frac{7}{2})\) 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 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( 1 + 11 p^{2} T^{2} + p^{10} T^{4} \)
5$C_2$ \( ( 1 - 98 T + p^{5} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 22566 T^{2} + p^{10} T^{4} \)
13$C_2^2$ \( 1 - 139414 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 + 2733134 T^{2} + p^{10} T^{4} \)
19$C_2^2$ \( 1 + 4764018 T^{2} + p^{10} T^{4} \)
23$C_2$ \( ( 1 + 376 T + p^{5} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 28445318 T^{2} + p^{10} T^{4} \)
31$C_2$ \( ( 1 - 120 p T + p^{5} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 9478 T + p^{5} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 176134622 T^{2} + p^{10} T^{4} \)
43$C_2^2$ \( 1 + 257187906 T^{2} + p^{10} T^{4} \)
47$C_2$ \( ( 1 + 5188 T + p^{5} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 34494 T + p^{5} T^{2} )^{2} \)
59$C_2$ \( ( 1 - 16440 T + p^{5} T^{2} )^{2} \)
61$C_2^2$ \( 1 - 676554918 T^{2} + p^{10} T^{4} \)
67$C_2$ \( ( 1 - 21108 T + p^{5} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 54868 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 3964459266 T^{2} + p^{10} T^{4} \)
79$C_2^2$ \( 1 + 4409774378 T^{2} + p^{10} T^{4} \)
83$C_2^2$ \( 1 + 2290925366 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 5666 T + p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 25918 T + p^{5} T^{2} )^{2} \)
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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

−9.479000060373768567343425873833, −8.959192562898767897786755877120, −8.635245550641465472997460539314, −8.556790062121901207759158050997, −7.83303908485447745784601758259, −6.94429388409575589559211121874, −6.63014088715510916553658933728, −6.51986801079625998547551668538, −5.64493878972093699895564426126, −5.54391782518456289370839016366, −5.15245401759655305241939194464, −5.02495370210870154421787724428, −4.06230335400468709019575152421, −3.82618828049690148957458772422, −2.87205604029316889919195147131, −2.35688786142741493962858122828, −2.19226454704284718301941525278, −1.44597950297667838953202204456, −1.04654805594445991641113152321, −0.50487426121503190800112073945, 0.50487426121503190800112073945, 1.04654805594445991641113152321, 1.44597950297667838953202204456, 2.19226454704284718301941525278, 2.35688786142741493962858122828, 2.87205604029316889919195147131, 3.82618828049690148957458772422, 4.06230335400468709019575152421, 5.02495370210870154421787724428, 5.15245401759655305241939194464, 5.54391782518456289370839016366, 5.64493878972093699895564426126, 6.51986801079625998547551668538, 6.63014088715510916553658933728, 6.94429388409575589559211121874, 7.83303908485447745784601758259, 8.556790062121901207759158050997, 8.635245550641465472997460539314, 8.959192562898767897786755877120, 9.479000060373768567343425873833

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