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Properties

Label 4-1210e2-1.1-c1e2-0-3
Degree $4$
Conductor $1464100$
Sign $1$
Analytic cond. $93.3522$
Root an. cond. $3.10835$
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  − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 3·7-s − 4·8-s + 2·9-s + 4·10-s + 6·12-s + 13-s + 6·14-s − 4·15-s + 5·16-s + 6·17-s − 4·18-s − 3·19-s − 6·20-s − 6·21-s + 7·23-s − 8·24-s + 3·25-s − 2·26-s + 6·27-s − 9·28-s − 2·29-s + 8·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.13·7-s − 1.41·8-s + 2/3·9-s + 1.26·10-s + 1.73·12-s + 0.277·13-s + 1.60·14-s − 1.03·15-s + 5/4·16-s + 1.45·17-s − 0.942·18-s − 0.688·19-s − 1.34·20-s − 1.30·21-s + 1.45·23-s − 1.63·24-s + 3/5·25-s − 0.392·26-s + 1.15·27-s − 1.70·28-s − 0.371·29-s + 1.46·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1464100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1464100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1464100\)    =    \(2^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(93.3522\)
Root analytic conductor: \(3.10835\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1464100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.228163024\)
\(L(\frac12)\) \(\approx\) \(1.228163024\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
7$D_{4}$ \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_p
13$D_{4}$ \( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_z
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bm
19$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_j
23$D_{4}$ \( 1 - 7 T + 47 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.23.ah_bv
29$D_{4}$ \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_cc
31$D_{4}$ \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_de
37$D_{4}$ \( 1 - 5 T + 69 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.37.af_cr
41$D_{4}$ \( 1 + 3 T + 73 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_cv
43$D_{4}$ \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.43.ao_fa
47$D_{4}$ \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.47.ah_bt
53$D_{4}$ \( 1 - 13 T + 117 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.53.an_en
59$D_{4}$ \( 1 - 7 T + 129 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.59.ah_ez
61$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.61.a_bq
67$D_{4}$ \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_fi
71$D_{4}$ \( 1 - 22 T + 258 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.71.aw_jy
73$D_{4}$ \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.73.k_ew
79$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_gs
83$D_{4}$ \( 1 + 18 T + 202 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_hu
89$D_{4}$ \( 1 - 5 T - 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.89.af_abb
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.97.y_na
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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

−9.753934297297095681413146683450, −9.485546546344515658303031623540, −9.020117914018887128852309870897, −8.823326044063476464685607206393, −8.270601874588487330773693815833, −8.154282609039941413717392264995, −7.51903847205411447776543352266, −7.21951551124691759693812085119, −6.95337259877675577187907734752, −6.53283270907535514896398735374, −5.66494433196878283870052222158, −5.66022680726739255361377533399, −4.70675088327309224806476965137, −3.99980373744748919123389781390, −3.47431637875756302287885451120, −3.34326090201067843302575779225, −2.60632031089986872035062694993, −2.28183430013606027378851758661, −1.20878608321599233149146862544, −0.63009409150402513985240828487, 0.63009409150402513985240828487, 1.20878608321599233149146862544, 2.28183430013606027378851758661, 2.60632031089986872035062694993, 3.34326090201067843302575779225, 3.47431637875756302287885451120, 3.99980373744748919123389781390, 4.70675088327309224806476965137, 5.66022680726739255361377533399, 5.66494433196878283870052222158, 6.53283270907535514896398735374, 6.95337259877675577187907734752, 7.21951551124691759693812085119, 7.51903847205411447776543352266, 8.154282609039941413717392264995, 8.270601874588487330773693815833, 8.823326044063476464685607206393, 9.020117914018887128852309870897, 9.485546546344515658303031623540, 9.753934297297095681413146683450

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