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Properties

Label 4-2106e2-1.1-c1e2-0-37
Degree $4$
Conductor $4435236$
Sign $1$
Analytic cond. $282.794$
Root an. cond. $4.10079$
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 + 3·4-s − 2·5-s − 4·7-s + 4·8-s − 4·10-s − 4·11-s − 2·13-s − 8·14-s + 5·16-s + 6·19-s − 6·20-s − 8·22-s − 2·23-s − 4·25-s − 4·26-s − 12·28-s − 2·29-s − 4·31-s + 6·32-s + 8·35-s − 12·37-s + 12·38-s − 8·40-s − 18·41-s − 4·43-s − 12·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.51·7-s + 1.41·8-s − 1.26·10-s − 1.20·11-s − 0.554·13-s − 2.13·14-s + 5/4·16-s + 1.37·19-s − 1.34·20-s − 1.70·22-s − 0.417·23-s − 4/5·25-s − 0.784·26-s − 2.26·28-s − 0.371·29-s − 0.718·31-s + 1.06·32-s + 1.35·35-s − 1.97·37-s + 1.94·38-s − 1.26·40-s − 2.81·41-s − 0.609·43-s − 1.80·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4435236\)    =    \(2^{2} \cdot 3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(282.794\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4435236,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_i
7$D_{4}$ \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_p
11$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_x
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_bj
23$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_u
29$D_{4}$ \( 1 + 2 T + 47 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_bv
31$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_cc
37$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_du
41$D_{4}$ \( 1 + 18 T + 160 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.41.s_ge
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_bq
47$D_{4}$ \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.47.q_fq
53$D_{4}$ \( 1 + 18 T + 175 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.53.s_gt
59$D_{4}$ \( 1 + 8 T + 131 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_fb
61$D_{4}$ \( 1 - 8 T + 111 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_eh
67$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_bq
71$D_{4}$ \( 1 + 10 T + 155 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.71.k_fz
73$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_fi
79$D_{4}$ \( 1 + 2 T + 132 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_fc
83$C_2^2$ \( 1 + 163 T^{2} + p^{2} T^{4} \) 2.83.a_gh
89$D_{4}$ \( 1 + 6 T + 160 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_ge
97$D_{4}$ \( 1 - 10 T + 216 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_ii
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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

−8.791827720066956787144312068013, −8.390444427887630571640885940820, −7.891191865697225727891874911811, −7.72631866169123943631201932539, −7.18593088857684276451753428607, −6.95215657817845486665494552397, −6.36189585671489131292085073389, −6.32090013847116062268562388481, −5.40749726705661795969894848484, −5.37732281051159300999161447710, −4.89901461160539060648847913943, −4.55716565246915203434837579636, −3.74307763382326247324373832505, −3.45680866896735572553399382802, −3.11165869962736693240710981549, −3.06288125545901121212302253757, −1.90040085272481334899903322657, −1.75488285063184498479822252343, 0, 0, 1.75488285063184498479822252343, 1.90040085272481334899903322657, 3.06288125545901121212302253757, 3.11165869962736693240710981549, 3.45680866896735572553399382802, 3.74307763382326247324373832505, 4.55716565246915203434837579636, 4.89901461160539060648847913943, 5.37732281051159300999161447710, 5.40749726705661795969894848484, 6.32090013847116062268562388481, 6.36189585671489131292085073389, 6.95215657817845486665494552397, 7.18593088857684276451753428607, 7.72631866169123943631201932539, 7.891191865697225727891874911811, 8.390444427887630571640885940820, 8.791827720066956787144312068013

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