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Properties

Label 4-351e2-1.1-c1e2-0-16
Degree $4$
Conductor $123201$
Sign $1$
Analytic cond. $7.85540$
Root an. cond. $1.67414$
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  + 2-s + 5·5-s − 2·7-s + 8-s + 5·10-s + 11-s + 2·13-s − 2·14-s − 16-s + 5·17-s − 7·19-s + 22-s − 3·23-s + 12·25-s + 2·26-s + 8·29-s − 6·32-s + 5·34-s − 10·35-s − 6·37-s − 7·38-s + 5·40-s + 18·41-s + 5·43-s − 3·46-s + 9·47-s − 11·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.23·5-s − 0.755·7-s + 0.353·8-s + 1.58·10-s + 0.301·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s + 1.21·17-s − 1.60·19-s + 0.213·22-s − 0.625·23-s + 12/5·25-s + 0.392·26-s + 1.48·29-s − 1.06·32-s + 0.857·34-s − 1.69·35-s − 0.986·37-s − 1.13·38-s + 0.790·40-s + 2.81·41-s + 0.762·43-s − 0.442·46-s + 1.31·47-s − 1.57·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(123201\)    =    \(3^{6} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(7.85540\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 123201,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.223888244\)
\(L(\frac12)\) \(\approx\) \(3.223888244\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_b
5$D_{4}$ \( 1 - p T + 13 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.5.af_n
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
11$D_{4}$ \( 1 - T + 19 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_t
17$D_{4}$ \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_bl
19$D_{4}$ \( 1 + 7 T + 47 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.19.h_bv
23$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_t
29$D_{4}$ \( 1 - 8 T + 61 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_cj
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_bf
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.41.as_gh
43$C_4$ \( 1 - 5 T + 89 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.43.af_dl
47$D_{4}$ \( 1 - 9 T + 85 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_dh
53$D_{4}$ \( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.53.p_fd
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.59.ag_ex
61$D_{4}$ \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_fp
67$D_{4}$ \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.67.j_cv
71$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_bi
73$D_{4}$ \( 1 + 11 T + 147 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_fr
79$D_{4}$ \( 1 + 17 T + 201 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.79.r_ht
83$D_{4}$ \( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_gn
89$D_{4}$ \( 1 - 10 T + 190 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_hi
97$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.97.a_ao
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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

−11.87881010773848584604510109492, −10.96632760512361187435883141703, −10.86207915658704134277858263223, −10.23686429166615985935362348652, −9.816225623354839980054513278094, −9.657874541498972227081295703374, −8.958084264753605810503780161478, −8.684436462320449432023153210425, −7.983282360637784219511460652492, −7.26090966157808981618460348382, −6.64389701921348176929445455401, −6.25161766782969212506130360366, −5.78398706958049095128073790077, −5.64807113852620899683647244577, −4.67324741992451976949119887348, −4.32979182289466911112539596343, −3.52832995184545241146778429788, −2.70744784226701525052801596168, −2.12516934195810893521729824153, −1.31726451116100674971788723911, 1.31726451116100674971788723911, 2.12516934195810893521729824153, 2.70744784226701525052801596168, 3.52832995184545241146778429788, 4.32979182289466911112539596343, 4.67324741992451976949119887348, 5.64807113852620899683647244577, 5.78398706958049095128073790077, 6.25161766782969212506130360366, 6.64389701921348176929445455401, 7.26090966157808981618460348382, 7.983282360637784219511460652492, 8.684436462320449432023153210425, 8.958084264753605810503780161478, 9.657874541498972227081295703374, 9.816225623354839980054513278094, 10.23686429166615985935362348652, 10.86207915658704134277858263223, 10.96632760512361187435883141703, 11.87881010773848584604510109492

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