Properties

Label 4-33e4-1.1-c2e2-0-2
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $880.492$
Root an. cond. $5.44730$
Motivic weight $2$
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  + 6·4-s + 14·5-s + 20·16-s + 84·20-s + 18·23-s + 97·25-s + 98·31-s + 34·37-s − 64·47-s + 48·49-s − 32·53-s + 142·59-s + 24·64-s − 62·67-s + 146·71-s + 280·80-s + 18·89-s + 108·92-s − 34·97-s + 582·100-s − 32·103-s − 130·113-s + 252·115-s + 588·124-s + 322·125-s + 127-s + 131-s + ⋯
L(s)  = 1  + 3/2·4-s + 14/5·5-s + 5/4·16-s + 21/5·20-s + 0.782·23-s + 3.87·25-s + 3.16·31-s + 0.918·37-s − 1.36·47-s + 0.979·49-s − 0.603·53-s + 2.40·59-s + 3/8·64-s − 0.925·67-s + 2.05·71-s + 7/2·80-s + 0.202·89-s + 1.17·92-s − 0.350·97-s + 5.81·100-s − 0.310·103-s − 1.15·113-s + 2.19·115-s + 4.74·124-s + 2.57·125-s + 0.00787·127-s + 0.00763·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(880.492\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(10.06441244\)
\(L(\frac12)\) \(\approx\) \(10.06441244\)
\(L(2)\) 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$C_2^2$ \( 1 - 3 p T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 - 7 T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 48 T^{2} + p^{4} T^{4} \)
13$C_2^2$ \( 1 - 50 T^{2} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 560 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \)
23$C_2$ \( ( 1 - 9 T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 1170 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 49 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 17 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3074 T^{2} + p^{4} T^{4} \)
43$C_2^2$ \( 1 - 1520 T^{2} + p^{4} T^{4} \)
47$C_2$ \( ( 1 + 32 T + p^{2} T^{2} )^{2} \)
53$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
59$C_2$ \( ( 1 - 71 T + p^{2} T^{2} )^{2} \)
61$C_2^2$ \( 1 - 7314 T^{2} + p^{4} T^{4} \)
67$C_2$ \( ( 1 + 31 T + p^{2} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 73 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 9090 T^{2} + p^{4} T^{4} \)
79$C_2^2$ \( 1 + 12160 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 12528 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 9 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 17 T + p^{2} T^{2} )^{2} \)
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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

−10.01524948606186479175692658378, −9.558296022145191707193031347545, −9.256209726012070296190809186602, −8.757519220548791085532378309870, −8.137259849319320112878156432762, −7.907600424657050321692870910694, −7.14840813063617745062015911594, −6.67810767473185864332906155709, −6.40052770563603359970333867415, −6.33578237492520727696628073616, −5.66315834758724107405792899010, −5.40518828495700970659082325800, −4.89536993201924350298564260728, −4.30410947364597596732599470767, −3.35441212460530370623727187718, −2.77278635843722614241562168013, −2.28336081677831405515085277785, −2.28305346052858889459469012778, −1.29623071465634502797093564536, −1.07775334221641359095138732183, 1.07775334221641359095138732183, 1.29623071465634502797093564536, 2.28305346052858889459469012778, 2.28336081677831405515085277785, 2.77278635843722614241562168013, 3.35441212460530370623727187718, 4.30410947364597596732599470767, 4.89536993201924350298564260728, 5.40518828495700970659082325800, 5.66315834758724107405792899010, 6.33578237492520727696628073616, 6.40052770563603359970333867415, 6.67810767473185864332906155709, 7.14840813063617745062015911594, 7.907600424657050321692870910694, 8.137259849319320112878156432762, 8.757519220548791085532378309870, 9.256209726012070296190809186602, 9.558296022145191707193031347545, 10.01524948606186479175692658378

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