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Properties

Label 4-351e2-1.1-c1e2-0-30
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 4·5-s + 7-s − 4·8-s + 8·10-s − 2·11-s − 7·13-s − 2·14-s + 8·16-s − 4·17-s − 3·19-s − 8·20-s + 4·22-s + 6·23-s + 2·25-s + 14·26-s + 2·28-s − 4·29-s − 10·31-s − 8·32-s + 8·34-s − 4·35-s − 7·37-s + 6·38-s + 16·40-s + 6·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.78·5-s + 0.377·7-s − 1.41·8-s + 2.52·10-s − 0.603·11-s − 1.94·13-s − 0.534·14-s + 2·16-s − 0.970·17-s − 0.688·19-s − 1.78·20-s + 0.852·22-s + 1.25·23-s + 2/5·25-s + 2.74·26-s + 0.377·28-s − 0.742·29-s − 1.79·31-s − 1.41·32-s + 1.37·34-s − 0.676·35-s − 1.15·37-s + 0.973·38-s + 2.52·40-s + 0.937·41-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: \(2\)
Selberg data: \((4,\ 123201,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 7 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_c
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_ab
19$C_2^2$ \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_ak
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_an
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.37.h_m
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.43.ad_abi
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.61.ao_ff
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.83.u_kg
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_ed
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−11.23929147779686842114946259679, −11.01506990538639655317233415465, −10.04927701558911797147737260933, −9.999655599652062749354586337855, −9.130026131790274782265772119627, −9.048696869156015434212847021671, −8.356586101532081295465464597177, −8.145477263238536082348213180195, −7.37157747022370287708577472595, −7.30780785495269611010069047714, −6.96366518931557363759223304398, −5.83535335045190574463888569713, −5.48023155938879234702754389777, −4.54273296193659099197143339155, −4.25686797473206717732428116321, −3.30216815333986900977104981674, −2.75066570718911401042078896995, −1.84453241696011185893081113645, 0, 0, 1.84453241696011185893081113645, 2.75066570718911401042078896995, 3.30216815333986900977104981674, 4.25686797473206717732428116321, 4.54273296193659099197143339155, 5.48023155938879234702754389777, 5.83535335045190574463888569713, 6.96366518931557363759223304398, 7.30780785495269611010069047714, 7.37157747022370287708577472595, 8.145477263238536082348213180195, 8.356586101532081295465464597177, 9.048696869156015434212847021671, 9.130026131790274782265772119627, 9.999655599652062749354586337855, 10.04927701558911797147737260933, 11.01506990538639655317233415465, 11.23929147779686842114946259679

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