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Properties

Label 4-219700-1.1-c1e2-0-7
Degree $4$
Conductor $219700$
Sign $1$
Analytic cond. $14.0082$
Root an. cond. $1.93462$
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  + 2·2-s + 3·4-s − 5-s + 2·7-s + 4·8-s + 3·9-s − 2·10-s − 13-s + 4·14-s + 5·16-s + 6·18-s − 3·20-s − 4·25-s − 2·26-s + 6·28-s + 4·29-s + 6·32-s − 2·35-s + 9·36-s + 6·37-s − 4·40-s − 3·45-s + 26·47-s − 11·49-s − 8·50-s − 3·52-s + 8·56-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.447·5-s + 0.755·7-s + 1.41·8-s + 9-s − 0.632·10-s − 0.277·13-s + 1.06·14-s + 5/4·16-s + 1.41·18-s − 0.670·20-s − 4/5·25-s − 0.392·26-s + 1.13·28-s + 0.742·29-s + 1.06·32-s − 0.338·35-s + 3/2·36-s + 0.986·37-s − 0.632·40-s − 0.447·45-s + 3.79·47-s − 1.57·49-s − 1.13·50-s − 0.416·52-s + 1.06·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 219700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(219700\)    =    \(2^{2} \cdot 5^{2} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(14.0082\)
Root analytic conductor: \(1.93462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 219700,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.687099046\)
\(L(\frac12)\) \(\approx\) \(4.687099046\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + T + p T^{2} \)
13$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.7.ac_p
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.47.aba_kd
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.71.a_en
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−9.026911386413610630551375476230, −8.420872023290300609672133626170, −7.78618160341898961599148312743, −7.45985560667588965978165906144, −7.23568791400333437059684221061, −6.51632105010820945514209338703, −5.88908450915949561478136129218, −5.69499349255850374306805786509, −4.65916643268043458244706899373, −4.61153453491182141362175201622, −4.15748343893950010861474305754, −3.45971551695615743734182930904, −2.76910930427420910776332481273, −2.04502161572044176240278653055, −1.23311319217689188134168770571, 1.23311319217689188134168770571, 2.04502161572044176240278653055, 2.76910930427420910776332481273, 3.45971551695615743734182930904, 4.15748343893950010861474305754, 4.61153453491182141362175201622, 4.65916643268043458244706899373, 5.69499349255850374306805786509, 5.88908450915949561478136129218, 6.51632105010820945514209338703, 7.23568791400333437059684221061, 7.45985560667588965978165906144, 7.78618160341898961599148312743, 8.420872023290300609672133626170, 9.026911386413610630551375476230

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