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Properties

Label 4-666e2-1.1-c1e2-0-16
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 + 3·5-s + 4·7-s + 8-s − 3·10-s + 12·11-s − 2·13-s − 4·14-s − 16-s + 3·17-s − 2·19-s − 12·22-s − 12·23-s + 5·25-s + 2·26-s − 6·29-s + 4·31-s − 3·34-s + 12·35-s + 11·37-s + 2·38-s + 3·40-s + 3·41-s − 8·43-s + 12·46-s + 12·47-s + 7·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.34·5-s + 1.51·7-s + 0.353·8-s − 0.948·10-s + 3.61·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 0.727·17-s − 0.458·19-s − 2.55·22-s − 2.50·23-s + 25-s + 0.392·26-s − 1.11·29-s + 0.718·31-s − 0.514·34-s + 2.02·35-s + 1.80·37-s + 0.324·38-s + 0.474·40-s + 0.468·41-s − 1.21·43-s + 1.76·46-s + 1.75·47-s + 49-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

\(L(1)\) \(\approx\) \(2.490691646\)
\(L(\frac12)\) \(\approx\) \(2.490691646\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_2$ \( 1 - 11 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ae_j
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.ab_aci
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2^2$ \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_cv
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_en
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_abv
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.89.ad_adc
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.97.ba_nz
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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

−10.64307399443433976726192149390, −10.10902638899157411483228433622, −9.664678436761957527124219106579, −9.521764319562659199178245474575, −9.127107215105970828885280331669, −8.690173801416819714488735108548, −8.157508080426453968141298082709, −7.901327502592385876726466737549, −7.17025985439968865239515539619, −6.81126569717800870079943760346, −6.22607661180026523194009299268, −5.76002121126875637677185898293, −5.65873858784129263721906016563, −4.47032060414079140953111468613, −4.27080625255500743610972926774, −4.00965329015128449021539138435, −2.90142315331319424496889821231, −1.79613906679756125345681931643, −1.75341270264198906883547249232, −1.12471837199076816930995517139, 1.12471837199076816930995517139, 1.75341270264198906883547249232, 1.79613906679756125345681931643, 2.90142315331319424496889821231, 4.00965329015128449021539138435, 4.27080625255500743610972926774, 4.47032060414079140953111468613, 5.65873858784129263721906016563, 5.76002121126875637677185898293, 6.22607661180026523194009299268, 6.81126569717800870079943760346, 7.17025985439968865239515539619, 7.901327502592385876726466737549, 8.157508080426453968141298082709, 8.690173801416819714488735108548, 9.127107215105970828885280331669, 9.521764319562659199178245474575, 9.664678436761957527124219106579, 10.10902638899157411483228433622, 10.64307399443433976726192149390

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