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Properties

Label 4-418176-1.1-c1e2-0-34
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 4·5-s + 6-s − 8-s + 9-s + 4·10-s − 12-s + 4·15-s + 16-s − 18-s + 8·19-s − 4·20-s − 8·23-s + 24-s + 2·25-s − 27-s − 12·29-s − 4·30-s − 32-s + 36-s − 8·38-s + 4·40-s + 8·43-s − 4·45-s + 8·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s + 1.03·15-s + 1/4·16-s − 0.235·18-s + 1.83·19-s − 0.894·20-s − 1.66·23-s + 0.204·24-s + 2/5·25-s − 0.192·27-s − 2.22·29-s − 0.730·30-s − 0.176·32-s + 1/6·36-s − 1.29·38-s + 0.632·40-s + 1.21·43-s − 0.596·45-s + 1.17·46-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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.a_acw
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
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$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−8.149492705426918418612295849228, −7.76149770434504125566929280206, −7.72333700605509306004633046648, −7.14262372141687320205238561411, −6.86624573821975866794841152731, −5.90428105253920776614679871565, −5.64980027395210574108319205224, −5.28231085552121772501988818495, −4.09502695849124324670131106925, −4.09304077381815838984743047096, −3.63086619141261654419365571100, −2.74090516225731871253746662847, −1.94521935316309572966615388289, −0.894868895370507138388762935569, 0, 0.894868895370507138388762935569, 1.94521935316309572966615388289, 2.74090516225731871253746662847, 3.63086619141261654419365571100, 4.09304077381815838984743047096, 4.09502695849124324670131106925, 5.28231085552121772501988818495, 5.64980027395210574108319205224, 5.90428105253920776614679871565, 6.86624573821975866794841152731, 7.14262372141687320205238561411, 7.72333700605509306004633046648, 7.76149770434504125566929280206, 8.149492705426918418612295849228

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