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Properties

Label 4-666e2-1.1-c1e2-0-27
Degree $4$
Conductor $443556$
Sign $1$
Analytic cond. $28.2815$
Root an. cond. $2.30608$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s + 4·5-s + 3·6-s − 8-s + 6·9-s + 4·10-s + 11-s − 6·13-s + 12·15-s − 16-s + 6·17-s + 6·18-s − 2·19-s + 22-s + 6·23-s − 3·24-s + 5·25-s − 6·26-s + 9·27-s + 12·30-s + 2·31-s + 3·33-s + 6·34-s + 2·37-s − 2·38-s − 18·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1.78·5-s + 1.22·6-s − 0.353·8-s + 2·9-s + 1.26·10-s + 0.301·11-s − 1.66·13-s + 3.09·15-s − 1/4·16-s + 1.45·17-s + 1.41·18-s − 0.458·19-s + 0.213·22-s + 1.25·23-s − 0.612·24-s + 25-s − 1.17·26-s + 1.73·27-s + 2.19·30-s + 0.359·31-s + 0.522·33-s + 1.02·34-s + 0.328·37-s − 0.324·38-s − 2.88·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(443556\)    =    \(2^{2} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(28.2815\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 443556,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.934999007\)
\(L(\frac12)\) \(\approx\) \(6.934999007\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_l
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_ak
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_abb
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.af_as
47$C_2^2$ \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_abf
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2^2$ \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.59.l_ck
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$C_2^2$ \( 1 + 15 T + 158 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.67.p_gc
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.73.k_gp
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_abr
83$C_2^2$ \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_acp
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 + 3 T - 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.97.d_adk
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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

−10.55082951457741611957993130786, −9.943595821880462257716174727871, −9.873016975895661991076048275966, −9.337688272681026416270595396808, −9.252241059919096688362242825717, −8.743354150476671103794894259464, −7.933934640929962433674461122829, −7.908991198085706424095052511726, −7.12217485441404289734932781738, −6.89855801716974714524573411143, −6.07381987054016463485060705327, −5.87841103485098055376736677230, −5.05572049751112669092680726517, −4.85724181469975152333303464075, −4.24925680893428164946973447359, −3.42131205082774140573758334602, −3.04116001306905430402924089523, −2.54565964853089336066968235778, −1.97478809787743220548822708940, −1.35402803091721531811731018178, 1.35402803091721531811731018178, 1.97478809787743220548822708940, 2.54565964853089336066968235778, 3.04116001306905430402924089523, 3.42131205082774140573758334602, 4.24925680893428164946973447359, 4.85724181469975152333303464075, 5.05572049751112669092680726517, 5.87841103485098055376736677230, 6.07381987054016463485060705327, 6.89855801716974714524573411143, 7.12217485441404289734932781738, 7.908991198085706424095052511726, 7.933934640929962433674461122829, 8.743354150476671103794894259464, 9.252241059919096688362242825717, 9.337688272681026416270595396808, 9.873016975895661991076048275966, 9.943595821880462257716174727871, 10.55082951457741611957993130786

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