Properties

Label 4-242e2-1.1-c1e2-0-4
Degree $4$
Conductor $58564$
Sign $1$
Analytic cond. $3.73409$
Root an. cond. $1.39010$
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 + 2·5-s + 6·6-s − 4·7-s + 4·8-s + 2·9-s + 4·10-s + 9·12-s − 2·13-s − 8·14-s + 6·15-s + 5·16-s + 17-s + 4·18-s − 5·19-s + 6·20-s − 12·21-s − 2·23-s + 12·24-s − 2·25-s − 4·26-s − 6·27-s − 12·28-s + 12·30-s + 4·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 3/2·4-s + 0.894·5-s + 2.44·6-s − 1.51·7-s + 1.41·8-s + 2/3·9-s + 1.26·10-s + 2.59·12-s − 0.554·13-s − 2.13·14-s + 1.54·15-s + 5/4·16-s + 0.242·17-s + 0.942·18-s − 1.14·19-s + 1.34·20-s − 2.61·21-s − 0.417·23-s + 2.44·24-s − 2/5·25-s − 0.784·26-s − 1.15·27-s − 2.26·28-s + 2.19·30-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(58564\)    =    \(2^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3.73409\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 58564,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.077968449\)
\(L(\frac12)\) \(\approx\) \(5.077968449\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$D_{4}$ \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.3.ad_h
5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_g
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_w
17$D_{4}$ \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_bh
19$D_{4}$ \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_bh
23$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bq
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_bm
41$D_{4}$ \( 1 - 9 T + 101 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_dx
43$D_{4}$ \( 1 - 3 T - 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.43.ad_an
47$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_da
53$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_es
59$D_{4}$ \( 1 + 15 T + 173 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.59.p_gr
61$D_{4}$ \( 1 - 4 T + 106 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_ec
67$D_{4}$ \( 1 - 11 T + 133 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.67.al_fd
71$D_{4}$ \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_fq
73$D_{4}$ \( 1 - 23 T + 277 T^{2} - 23 p T^{3} + p^{2} T^{4} \) 2.73.ax_kr
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$D_{4}$ \( 1 - 3 T + 107 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.83.ad_ed
89$D_{4}$ \( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.89.f_fx
97$D_{4}$ \( 1 - 21 T + 293 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.97.av_lh
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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

−12.45618112546636037692280963740, −12.41591394572359680257222272584, −11.36292163946739514071372102626, −11.14308816516032217922027637895, −10.23988494776451817843304961024, −9.932749541309653048024732098464, −9.296910195920915486863152315656, −9.292889371651074355920669472020, −8.312168975313695718293713980838, −7.925476634561000018722407887335, −7.42742997818109701165262739129, −6.54097981734495754034534956685, −6.12040232995040826334216587585, −6.00283944129234321138458958830, −4.94563136271973363431212875724, −4.35330270706399664727065628455, −3.40998762431223147088630745347, −3.28338095505031098489531796514, −2.39534755043460532782145014653, −2.15746909252528951425940770847, 2.15746909252528951425940770847, 2.39534755043460532782145014653, 3.28338095505031098489531796514, 3.40998762431223147088630745347, 4.35330270706399664727065628455, 4.94563136271973363431212875724, 6.00283944129234321138458958830, 6.12040232995040826334216587585, 6.54097981734495754034534956685, 7.42742997818109701165262739129, 7.925476634561000018722407887335, 8.312168975313695718293713980838, 9.292889371651074355920669472020, 9.296910195920915486863152315656, 9.932749541309653048024732098464, 10.23988494776451817843304961024, 11.14308816516032217922027637895, 11.36292163946739514071372102626, 12.41591394572359680257222272584, 12.45618112546636037692280963740

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