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Properties

Label 4-8954e2-1.1-c1e2-0-3
Degree $4$
Conductor $80174116$
Sign $1$
Analytic cond. $5111.97$
Root an. cond. $8.45565$
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 + 3·3-s + 3·4-s − 5-s + 6·6-s − 2·7-s + 4·8-s + 4·9-s − 2·10-s + 9·12-s + 13-s − 4·14-s − 3·15-s + 5·16-s + 12·17-s + 8·18-s − 4·19-s − 3·20-s − 6·21-s − 3·23-s + 12·24-s − 6·25-s + 2·26-s + 6·27-s − 6·28-s − 3·29-s − 6·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 3/2·4-s − 0.447·5-s + 2.44·6-s − 0.755·7-s + 1.41·8-s + 4/3·9-s − 0.632·10-s + 2.59·12-s + 0.277·13-s − 1.06·14-s − 0.774·15-s + 5/4·16-s + 2.91·17-s + 1.88·18-s − 0.917·19-s − 0.670·20-s − 1.30·21-s − 0.625·23-s + 2.44·24-s − 6/5·25-s + 0.392·26-s + 1.15·27-s − 1.13·28-s − 0.557·29-s − 1.09·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80174116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80174116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(80174116\)    =    \(2^{2} \cdot 11^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(5111.97\)
Root analytic conductor: \(8.45565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 80174116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(15.96064518\)
\(L(\frac12)\) \(\approx\) \(15.96064518\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good3$C_4$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.3.ad_f
5$D_{4}$ \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_h
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_c
13$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_x
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_t
29$C_4$ \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_bf
31$D_{4}$ \( 1 - 3 T + 61 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.31.ad_cj
41$D_{4}$ \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_cv
43$D_{4}$ \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_de
47$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_de
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_fy
61$D_{4}$ \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.61.ad_br
67$D_{4}$ \( 1 - 11 T + 83 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.67.al_df
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$D_{4}$ \( 1 - 21 T + 253 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.73.av_jt
79$D_{4}$ \( 1 - 7 T + 11 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.79.ah_l
83$D_{4}$ \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.83.u_ig
89$D_{4}$ \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_fa
97$D_{4}$ \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_ak
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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

−7.85127799871097651693780737745, −7.71157866799394495605253979055, −7.29225082054417778499650203958, −6.68551520258095914812797496216, −6.66479809651979962489050042649, −6.22465368323738475170226821087, −5.74790021373330847389479407776, −5.37334396147741940951367562840, −5.30287619962338141236602974563, −4.67009263454832620181363519777, −4.10654327673380068897849834263, −3.88893333963176893407111324264, −3.62114167542665222126625851513, −3.40621621974509071652066146448, −2.90931643311250936308460054015, −2.76289523818575905946027685980, −2.02374908954706742347164089517, −1.99135083638354144818915967395, −1.17718697634270251882419943489, −0.61726113112337577104791094293, 0.61726113112337577104791094293, 1.17718697634270251882419943489, 1.99135083638354144818915967395, 2.02374908954706742347164089517, 2.76289523818575905946027685980, 2.90931643311250936308460054015, 3.40621621974509071652066146448, 3.62114167542665222126625851513, 3.88893333963176893407111324264, 4.10654327673380068897849834263, 4.67009263454832620181363519777, 5.30287619962338141236602974563, 5.37334396147741940951367562840, 5.74790021373330847389479407776, 6.22465368323738475170226821087, 6.66479809651979962489050042649, 6.68551520258095914812797496216, 7.29225082054417778499650203958, 7.71157866799394495605253979055, 7.85127799871097651693780737745

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