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Properties

Label 4-5766e2-1.1-c1e2-0-2
Degree $4$
Conductor $33246756$
Sign $1$
Analytic cond. $2119.84$
Root an. cond. $6.78540$
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 − 6·5-s + 4·6-s + 4·7-s − 4·8-s + 3·9-s + 12·10-s + 6·11-s − 6·12-s − 6·13-s − 8·14-s + 12·15-s + 5·16-s − 6·17-s − 6·18-s + 2·19-s − 18·20-s − 8·21-s − 12·22-s + 8·23-s + 8·24-s + 19·25-s + 12·26-s − 4·27-s + 12·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s − 2.68·5-s + 1.63·6-s + 1.51·7-s − 1.41·8-s + 9-s + 3.79·10-s + 1.80·11-s − 1.73·12-s − 1.66·13-s − 2.13·14-s + 3.09·15-s + 5/4·16-s − 1.45·17-s − 1.41·18-s + 0.458·19-s − 4.02·20-s − 1.74·21-s − 2.55·22-s + 1.66·23-s + 1.63·24-s + 19/5·25-s + 2.35·26-s − 0.769·27-s + 2.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33246756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33246756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33246756\)    =    \(2^{2} \cdot 3^{2} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(2119.84\)
Root analytic conductor: \(6.78540\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 33246756,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5787533031\)
\(L(\frac12)\) \(\approx\) \(0.5787533031\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
31 \( 1 \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.5.g_r
7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$D_{4}$ \( 1 + 6 T + 33 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_bh
17$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bj
19$D_{4}$ \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_h
23$D_{4}$ \( 1 - 8 T + 44 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bs
29$D_{4}$ \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_cu
37$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_de
41$D_{4}$ \( 1 + 8 T + 80 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_dc
43$D_{4}$ \( 1 - 16 T + 148 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.43.aq_fs
47$D_{4}$ \( 1 + 10 T + 117 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_en
53$D_{4}$ \( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_eu
59$D_{4}$ \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_by
61$D_{4}$ \( 1 + 10 T + 145 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_fp
67$D_{4}$ \( 1 - 14 T + 151 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_fv
71$D_{4}$ \( 1 + 2 T + 125 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_ev
73$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \) 2.73.a_bw
79$D_{4}$ \( 1 + 2 T + 61 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_cj
83$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_cl
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.89.a_fq
97$D_{4}$ \( 1 + 6 T + 195 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_hn
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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

−8.207748540636653322879326202863, −7.965256909295838743820199793751, −7.50381397018671123601698387380, −7.40887100255026011088594770145, −6.93401259258691414846028618990, −6.85679994107058310326884933383, −6.31436120657170730850553022155, −6.13367014181589791569156030602, −5.15216959442702293203992330905, −4.94569752888614640859071495219, −4.63802972020580127139219304120, −4.43281771143788091713413696267, −3.95045181729761275550677621635, −3.58222450740527879032291245861, −2.77958626865337149500515345478, −2.68928823137708934335773454499, −1.54478722590053019192525653082, −1.52068371807264032003466782898, −0.63162444083299079666649523901, −0.49188365232248440895710325323, 0.49188365232248440895710325323, 0.63162444083299079666649523901, 1.52068371807264032003466782898, 1.54478722590053019192525653082, 2.68928823137708934335773454499, 2.77958626865337149500515345478, 3.58222450740527879032291245861, 3.95045181729761275550677621635, 4.43281771143788091713413696267, 4.63802972020580127139219304120, 4.94569752888614640859071495219, 5.15216959442702293203992330905, 6.13367014181589791569156030602, 6.31436120657170730850553022155, 6.85679994107058310326884933383, 6.93401259258691414846028618990, 7.40887100255026011088594770145, 7.50381397018671123601698387380, 7.965256909295838743820199793751, 8.207748540636653322879326202863

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