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Properties

Label 4-351e2-1.1-c1e2-0-13
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  + 2·4-s + 7-s + 5·13-s + 19-s − 10·25-s + 2·28-s + 22·31-s + 37-s − 5·43-s + 7·49-s + 10·52-s − 14·61-s − 8·64-s + 16·67-s − 20·73-s + 2·76-s − 8·79-s + 5·91-s − 14·97-s − 20·100-s − 26·103-s + 34·109-s + 11·121-s + 44·124-s + 127-s + 131-s + 133-s + ⋯
L(s)  = 1  + 4-s + 0.377·7-s + 1.38·13-s + 0.229·19-s − 2·25-s + 0.377·28-s + 3.95·31-s + 0.164·37-s − 0.762·43-s + 49-s + 1.38·52-s − 1.79·61-s − 64-s + 1.95·67-s − 2.34·73-s + 0.229·76-s − 0.900·79-s + 0.524·91-s − 1.42·97-s − 2·100-s − 2.56·103-s + 3.25·109-s + 121-s + 3.95·124-s + 0.0887·127-s + 0.0873·131-s + 0.0867·133-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

\(L(1)\) \(\approx\) \(2.258013812\)
\(L(\frac12)\) \(\approx\) \(2.258013812\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 5 T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.31.aw_hb
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.f_as
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.67.aq_hh
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.71.a_act
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.o_dv
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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

−11.72994900794972896813416929985, −11.25676899997015512880344279746, −10.93801442923165363108391144313, −10.39902513649918227290104549305, −9.759136078445770395785699030573, −9.751948866369155556987821554001, −8.642136421872789978466293635967, −8.499155206088973980616231500213, −7.974764320598132063965579275233, −7.49308654997480279322648289931, −6.86787480392456193785976295392, −6.39123393806874710010754111298, −5.98276939554094889992622433913, −5.58790945103634883965906037079, −4.47364420962004874604194255273, −4.36099563197412917962413937700, −3.35792346327736903891873731284, −2.78346195967315798779579267712, −1.99739779454235381778313161050, −1.16850586335834871646812771678, 1.16850586335834871646812771678, 1.99739779454235381778313161050, 2.78346195967315798779579267712, 3.35792346327736903891873731284, 4.36099563197412917962413937700, 4.47364420962004874604194255273, 5.58790945103634883965906037079, 5.98276939554094889992622433913, 6.39123393806874710010754111298, 6.86787480392456193785976295392, 7.49308654997480279322648289931, 7.974764320598132063965579275233, 8.499155206088973980616231500213, 8.642136421872789978466293635967, 9.751948866369155556987821554001, 9.759136078445770395785699030573, 10.39902513649918227290104549305, 10.93801442923165363108391144313, 11.25676899997015512880344279746, 11.72994900794972896813416929985

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