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Properties

Label 4-380e2-1.1-c1e2-0-4
Degree $4$
Conductor $144400$
Sign $1$
Analytic cond. $9.20706$
Root an. cond. $1.74192$
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 + 2·3-s + 2·4-s − 4·5-s + 4·6-s + 4·7-s + 2·9-s − 8·10-s + 4·12-s + 8·14-s − 8·15-s − 4·16-s + 10·17-s + 4·18-s + 2·19-s − 8·20-s + 8·21-s − 8·23-s + 11·25-s + 6·27-s + 8·28-s − 16·30-s − 8·32-s + 20·34-s − 16·35-s + 4·36-s + 4·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s + 1.51·7-s + 2/3·9-s − 2.52·10-s + 1.15·12-s + 2.13·14-s − 2.06·15-s − 16-s + 2.42·17-s + 0.942·18-s + 0.458·19-s − 1.78·20-s + 1.74·21-s − 1.66·23-s + 11/5·25-s + 1.15·27-s + 1.51·28-s − 2.92·30-s − 1.41·32-s + 3.42·34-s − 2.70·35-s + 2/3·36-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(144400\)    =    \(2^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(9.20706\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 144400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.413846429\)
\(L(\frac12)\) \(\approx\) \(4.413846429\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_i
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2^2$ \( 1 + p^{2} T^{4} \) 2.13.a_a
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ak_by
23$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_bg
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2^2$ \( 1 + p^{2} T^{4} \) 2.37.a_a
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_cu
47$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_i
53$C_2^2$ \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.53.u_hs
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.59.u_ik
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_s
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ak_by
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_bg
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2^2$ \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.97.u_hs
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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

−11.87636244189049786114796173309, −11.06583849387050060505520002028, −11.06286219608012253202174852197, −10.46692530324337044241769577694, −9.466364235530592190458860281079, −9.445663812620590631911760050741, −8.459582899911114216542190792991, −8.191806405495216471193949116886, −7.83868937167427018725984537670, −7.55959236752472271406399563547, −7.09909375133523476221715656395, −6.07806367325204372861851192403, −5.73999413700006855073118716641, −4.77815000529713457625027552916, −4.70908441683338009113550475769, −4.04665746341830002899512634653, −3.43713756132659844980657979210, −3.20506358796431622833557285978, −2.36203056334514511723485276137, −1.23780297017531345497225522510, 1.23780297017531345497225522510, 2.36203056334514511723485276137, 3.20506358796431622833557285978, 3.43713756132659844980657979210, 4.04665746341830002899512634653, 4.70908441683338009113550475769, 4.77815000529713457625027552916, 5.73999413700006855073118716641, 6.07806367325204372861851192403, 7.09909375133523476221715656395, 7.55959236752472271406399563547, 7.83868937167427018725984537670, 8.191806405495216471193949116886, 8.459582899911114216542190792991, 9.445663812620590631911760050741, 9.466364235530592190458860281079, 10.46692530324337044241769577694, 11.06286219608012253202174852197, 11.06583849387050060505520002028, 11.87636244189049786114796173309

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