Properties

Label 4-3698e2-1.1-c1e2-0-0
Degree $4$
Conductor $13675204$
Sign $1$
Analytic cond. $871.942$
Root an. cond. $5.43402$
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 − 3-s + 3·4-s + 3·5-s + 2·6-s − 4·8-s − 4·9-s − 6·10-s − 4·11-s − 3·12-s − 3·15-s + 5·16-s − 17-s + 8·18-s − 11·19-s + 9·20-s + 8·22-s + 3·23-s + 4·24-s − 2·25-s + 6·27-s + 7·29-s + 6·30-s + 13·31-s − 6·32-s + 4·33-s + 2·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 3/2·4-s + 1.34·5-s + 0.816·6-s − 1.41·8-s − 4/3·9-s − 1.89·10-s − 1.20·11-s − 0.866·12-s − 0.774·15-s + 5/4·16-s − 0.242·17-s + 1.88·18-s − 2.52·19-s + 2.01·20-s + 1.70·22-s + 0.625·23-s + 0.816·24-s − 2/5·25-s + 1.15·27-s + 1.29·29-s + 1.09·30-s + 2.33·31-s − 1.06·32-s + 0.696·33-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13675204 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13675204 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13675204\)    =    \(2^{2} \cdot 43^{4}\)
Sign: $1$
Analytic conductor: \(871.942\)
Root analytic conductor: \(5.43402\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13675204,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4854274774\)
\(L(\frac12)\) \(\approx\) \(0.4854274774\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
43 \( 1 \)
good3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_f
5$D_{4}$ \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_l
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_g
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$D_{4}$ \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) 2.17.b_bh
19$D_{4}$ \( 1 + 11 T + 67 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.19.l_cp
23$D_{4}$ \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_bl
29$D_{4}$ \( 1 - 7 T + 59 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.29.ah_ch
31$D_{4}$ \( 1 - 13 T + 103 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.31.an_dz
37$D_{4}$ \( 1 - 5 T + 79 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.37.af_db
41$D_{4}$ \( 1 + 5 T + 77 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.41.f_cz
47$D_{4}$ \( 1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_bj
53$D_{4}$ \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.53.k_ew
59$D_{4}$ \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.59.aq_gg
61$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_bu
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$D_{4}$ \( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.71.aq_he
73$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_cs
79$D_{4}$ \( 1 + T + 157 T^{2} + p T^{3} + p^{2} T^{4} \) 2.79.b_gb
83$D_{4}$ \( 1 + 10 T + 146 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_fq
89$D_{4}$ \( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_fe
97$D_{4}$ \( 1 + 11 T + 193 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.97.l_hl
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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

−8.508795676529503989571455323025, −8.446468627080632817038035861562, −8.125677595924454355072419992531, −7.947252821737219981375718899216, −7.15819917276149378004140169481, −6.79450123543375637019609730722, −6.41132919737417422811838814883, −6.25572375781201132898287390662, −5.79497256198037108778924707598, −5.67585148374969747185019754816, −4.89708912775427674778412318795, −4.86008512085474578454633745843, −4.15363814951323639130910445969, −3.50012471794639135822205989039, −2.65194026227083988740942261047, −2.54005758841697303092160920392, −2.41507429018567621460876591858, −1.69443957973565474575639402535, −0.952148985626179587589907303089, −0.31044157053758665361720108034, 0.31044157053758665361720108034, 0.952148985626179587589907303089, 1.69443957973565474575639402535, 2.41507429018567621460876591858, 2.54005758841697303092160920392, 2.65194026227083988740942261047, 3.50012471794639135822205989039, 4.15363814951323639130910445969, 4.86008512085474578454633745843, 4.89708912775427674778412318795, 5.67585148374969747185019754816, 5.79497256198037108778924707598, 6.25572375781201132898287390662, 6.41132919737417422811838814883, 6.79450123543375637019609730722, 7.15819917276149378004140169481, 7.947252821737219981375718899216, 8.125677595924454355072419992531, 8.446468627080632817038035861562, 8.508795676529503989571455323025

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