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Properties

Label 4-558e2-1.1-c1e2-0-6
Degree $4$
Conductor $311364$
Sign $1$
Analytic cond. $19.8528$
Root an. cond. $2.11084$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 2·5-s − 7-s − 4·8-s + 4·10-s + 3·11-s + 4·13-s + 2·14-s + 5·16-s + 8·19-s − 6·20-s − 6·22-s + 4·23-s + 5·25-s − 8·26-s − 3·28-s + 18·29-s − 7·31-s − 6·32-s + 2·35-s + 2·37-s − 16·38-s + 8·40-s + 2·41-s + 9·44-s − 8·46-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.377·7-s − 1.41·8-s + 1.26·10-s + 0.904·11-s + 1.10·13-s + 0.534·14-s + 5/4·16-s + 1.83·19-s − 1.34·20-s − 1.27·22-s + 0.834·23-s + 25-s − 1.56·26-s − 0.566·28-s + 3.34·29-s − 1.25·31-s − 1.06·32-s + 0.338·35-s + 0.328·37-s − 2.59·38-s + 1.26·40-s + 0.312·41-s + 1.35·44-s − 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(311364\)    =    \(2^{2} \cdot 3^{4} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(19.8528\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 311364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.015147887\)
\(L(\frac12)\) \(\approx\) \(1.015147887\)
\(L(\frac{3}{2})\) 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 \( 1 \)
31$C_2$ \( 1 + 7 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_ab
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.ai_bt
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.29.as_fj
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_abh
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_abl
43$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.43.a_abr
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$C_2^2$ \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \) 2.53.b_aca
59$C_2^2$ \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \) 2.59.ab_acg
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2^2$ \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_cv
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.73.ag_abl
79$C_2^2$ \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_cn
83$C_2^2$ \( 1 - 11 T + 38 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.83.al_bm
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_jj
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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

−10.74515051636514415587841598648, −10.55865111689719398676216737349, −10.20311498756124147961819501190, −9.466583105984742720365857735756, −9.135208178035983247041089573027, −8.946933576488088702254070736163, −8.259272003837856540113380774971, −8.143975239809663093280589616021, −7.34028183393403457838721613992, −7.18832553870333911869282176341, −6.45204493554862644475949901846, −6.44348412339406311741752888559, −5.49568356755595257404456565253, −5.06113903664733917129621822295, −4.11861701144546747821863020192, −3.69854973497502664321079861795, −3.03218867731255770971698441239, −2.55638730761401973659049897892, −1.08179301237228245172226052496, −1.01108969557264737248074254455, 1.01108969557264737248074254455, 1.08179301237228245172226052496, 2.55638730761401973659049897892, 3.03218867731255770971698441239, 3.69854973497502664321079861795, 4.11861701144546747821863020192, 5.06113903664733917129621822295, 5.49568356755595257404456565253, 6.44348412339406311741752888559, 6.45204493554862644475949901846, 7.18832553870333911869282176341, 7.34028183393403457838721613992, 8.143975239809663093280589616021, 8.259272003837856540113380774971, 8.946933576488088702254070736163, 9.135208178035983247041089573027, 9.466583105984742720365857735756, 10.20311498756124147961819501190, 10.55865111689719398676216737349, 10.74515051636514415587841598648

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