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Properties

Label 4-7360e2-1.1-c1e2-0-5
Degree $4$
Conductor $54169600$
Sign $1$
Analytic cond. $3453.90$
Root an. cond. $7.66615$
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-s − 2·5-s − 2·7-s − 9-s − 8·11-s − 3·13-s − 2·15-s + 2·17-s + 8·19-s − 2·21-s − 2·23-s + 3·25-s + 13·29-s − 7·31-s − 8·33-s + 4·35-s + 6·37-s − 3·39-s − 3·41-s − 10·43-s + 2·45-s − 5·47-s + 6·49-s + 2·51-s − 8·53-s + 16·55-s + 8·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.755·7-s − 1/3·9-s − 2.41·11-s − 0.832·13-s − 0.516·15-s + 0.485·17-s + 1.83·19-s − 0.436·21-s − 0.417·23-s + 3/5·25-s + 2.41·29-s − 1.25·31-s − 1.39·33-s + 0.676·35-s + 0.986·37-s − 0.480·39-s − 0.468·41-s − 1.52·43-s + 0.298·45-s − 0.729·47-s + 6/7·49-s + 0.280·51-s − 1.09·53-s + 2.15·55-s + 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(54169600\)    =    \(2^{12} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3453.90\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 54169600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.843226969\)
\(L(\frac12)\) \(\approx\) \(1.843226969\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ac
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_y
17$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_s
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
29$D_{4}$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.29.an_ds
31$D_{4}$ \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.31.h_cs
37$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_co
41$D_{4}$ \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_dc
43$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_dq
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_ck
53$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_cc
59$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_cc
61$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.61.o_fy
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_cs
71$D_{4}$ \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.71.af_eg
73$D_{4}$ \( 1 - 29 T + 352 T^{2} - 29 p T^{3} + p^{2} T^{4} \) 2.73.abd_no
79$D_{4}$ \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_dq
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.97.q_hi
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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

−8.019466183420974472587643588824, −7.920003987683617080633415380056, −7.36628249157273546848088507700, −7.31708656780234462144694769298, −6.68029375515146915962862185926, −6.48002015788764152785191309412, −5.98013427383560199043223379249, −5.52421130337419995200167818616, −5.05553287263736246179192832583, −4.93934495872797654627550755681, −4.80577167469353032857337036302, −4.02884001803698527951733214879, −3.50512903885484792693400758692, −3.26835051670116669468339029697, −2.91585641256777934365847666334, −2.74606438165345068045608314156, −2.21822531258096754953902037075, −1.65297435620234909557534245972, −0.60311664950922320690977588645, −0.52581791263379655264596093590, 0.52581791263379655264596093590, 0.60311664950922320690977588645, 1.65297435620234909557534245972, 2.21822531258096754953902037075, 2.74606438165345068045608314156, 2.91585641256777934365847666334, 3.26835051670116669468339029697, 3.50512903885484792693400758692, 4.02884001803698527951733214879, 4.80577167469353032857337036302, 4.93934495872797654627550755681, 5.05553287263736246179192832583, 5.52421130337419995200167818616, 5.98013427383560199043223379249, 6.48002015788764152785191309412, 6.68029375515146915962862185926, 7.31708656780234462144694769298, 7.36628249157273546848088507700, 7.920003987683617080633415380056, 8.019466183420974472587643588824

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