Properties

Label 4-4851e2-1.1-c1e2-0-1
Degree $4$
Conductor $23532201$
Sign $1$
Analytic cond. $1500.43$
Root an. cond. $6.22377$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4·4-s + 3·8-s + 2·11-s − 2·13-s + 3·16-s − 12·17-s − 4·19-s + 6·22-s − 4·23-s − 5·25-s − 6·26-s + 16·29-s + 8·31-s + 6·32-s − 36·34-s − 10·37-s − 12·38-s − 12·41-s + 8·43-s + 8·44-s − 12·46-s − 14·47-s − 15·50-s − 8·52-s + 8·53-s + 48·58-s + ⋯
L(s)  = 1  + 2.12·2-s + 2·4-s + 1.06·8-s + 0.603·11-s − 0.554·13-s + 3/4·16-s − 2.91·17-s − 0.917·19-s + 1.27·22-s − 0.834·23-s − 25-s − 1.17·26-s + 2.97·29-s + 1.43·31-s + 1.06·32-s − 6.17·34-s − 1.64·37-s − 1.94·38-s − 1.87·41-s + 1.21·43-s + 1.20·44-s − 1.76·46-s − 2.04·47-s − 2.12·50-s − 1.10·52-s + 1.09·53-s + 6.30·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23532201\)    =    \(3^{4} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1500.43\)
Root analytic conductor: \(6.22377\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23532201,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.688632290\)
\(L(\frac12)\) \(\approx\) \(3.688632290\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 16 T + 117 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−8.335681733955675014519856439721, −8.310433093170564276830090411001, −7.59895260696743776149588953182, −6.91439035429333203162134790866, −6.91062947084862944306200203169, −6.51114915634020253482846225053, −6.13706364626533386518532266294, −6.05835107238965807890654737834, −5.24235802543203367006570695931, −5.06275575734138551676589879296, −4.55843527103297888703111719756, −4.49928462116935228321695260150, −3.97262724535007170213709378782, −3.96830138490149366441009864577, −3.07673738124808432423060866294, −2.92705263726679144138433776591, −2.29174517526254193813568760293, −1.99657620414172471336668227116, −1.38579357828222635179628027076, −0.33620977718061315699853016858, 0.33620977718061315699853016858, 1.38579357828222635179628027076, 1.99657620414172471336668227116, 2.29174517526254193813568760293, 2.92705263726679144138433776591, 3.07673738124808432423060866294, 3.96830138490149366441009864577, 3.97262724535007170213709378782, 4.49928462116935228321695260150, 4.55843527103297888703111719756, 5.06275575734138551676589879296, 5.24235802543203367006570695931, 6.05835107238965807890654737834, 6.13706364626533386518532266294, 6.51114915634020253482846225053, 6.91062947084862944306200203169, 6.91439035429333203162134790866, 7.59895260696743776149588953182, 8.310433093170564276830090411001, 8.335681733955675014519856439721

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