↑

Properties

Label 4-418176-1.1-c1e2-0-54
Degree $4$
Conductor $418176$
Sign $-1$
Analytic cond. $26.6632$
Root an. cond. $2.27236$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 8·5-s + 6-s + 8-s + 9-s − 8·10-s + 12-s − 8·15-s + 16-s + 18-s − 8·20-s − 12·23-s + 24-s + 38·25-s + 27-s + 20·29-s − 8·30-s + 32-s + 36-s − 8·40-s + 8·43-s − 8·45-s − 12·46-s − 4·47-s + 48-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 3.57·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 2.52·10-s + 0.288·12-s − 2.06·15-s + 1/4·16-s + 0.235·18-s − 1.78·20-s − 2.50·23-s + 0.204·24-s + 38/5·25-s + 0.192·27-s + 3.71·29-s − 1.46·30-s + 0.176·32-s + 1/6·36-s − 1.26·40-s + 1.21·43-s − 1.19·45-s − 1.76·46-s − 0.583·47-s + 0.144·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(418176\)    =    \(2^{7} \cdot 3^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(26.6632\)
Root analytic conductor: \(2.27236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 418176,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3$C_1$ \( 1 - T \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.5.i_ba
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.29.au_gc
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.47.e_du
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.239518620901196099696917715149, −7.76468949407227890278331338099, −7.72339303053252541131209387915, −7.22439214684360967772263804706, −6.46917181705411800265373292912, −6.41339833010794299582499150983, −5.35095261869222727266089892863, −4.50249250702361606334840833954, −4.31442539583717353216143069554, −4.25842178547984493307347569445, −3.34833436263269527102959669443, −3.19368686523870503450682170613, −2.49691763110340121310810466350, −1.09804741173349039454069806686, 0, 1.09804741173349039454069806686, 2.49691763110340121310810466350, 3.19368686523870503450682170613, 3.34833436263269527102959669443, 4.25842178547984493307347569445, 4.31442539583717353216143069554, 4.50249250702361606334840833954, 5.35095261869222727266089892863, 6.41339833010794299582499150983, 6.46917181705411800265373292912, 7.22439214684360967772263804706, 7.72339303053252541131209387915, 7.76468949407227890278331338099, 8.239518620901196099696917715149

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