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Properties

Label 4-2178e2-1.1-c1e2-0-22
Degree $4$
Conductor $4743684$
Sign $1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
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 + 3·4-s − 2·5-s + 4·7-s + 4·8-s − 4·10-s + 2·13-s + 8·14-s + 5·16-s + 17-s + 5·19-s − 6·20-s + 2·23-s − 2·25-s + 4·26-s + 12·28-s + 4·31-s + 6·32-s + 2·34-s − 8·35-s + 6·37-s + 10·38-s − 8·40-s + 9·41-s − 3·43-s + 4·46-s + 4·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.51·7-s + 1.41·8-s − 1.26·10-s + 0.554·13-s + 2.13·14-s + 5/4·16-s + 0.242·17-s + 1.14·19-s − 1.34·20-s + 0.417·23-s − 2/5·25-s + 0.784·26-s + 2.26·28-s + 0.718·31-s + 1.06·32-s + 0.342·34-s − 1.35·35-s + 0.986·37-s + 1.62·38-s − 1.26·40-s + 1.40·41-s − 0.457·43-s + 0.589·46-s + 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4743684\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(302.461\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.463280749\)
\(L(\frac12)\) \(\approx\) \(8.463280749\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_g
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
13$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_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.af_bh
23$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_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.d_an
47$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_da
53$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_es
59$D_{4}$ \( 1 - 15 T + 173 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.59.ap_gr
61$D_{4}$ \( 1 + 4 T + 106 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_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.ag_fq
73$D_{4}$ \( 1 + 23 T + 277 T^{2} + 23 p T^{3} + p^{2} T^{4} \) 2.73.x_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.af_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

−9.037735255667995203201388236497, −8.869494181048331324556504621787, −8.190529888951037692011295760577, −8.039672198377512682103634579694, −7.53537913486923803595366408431, −7.50695737289222644858790861630, −6.88418751275801213484324672289, −6.49798999140538658481477969393, −5.84078569602366248045978915245, −5.72011169734652380728607285202, −5.11327606168201901615325305948, −4.89070470802441293896587460836, −4.24702578211883245021715298179, −4.22837792274623027778098214083, −3.45515698969463155512577447060, −3.34200650166558958121524680272, −2.44196075027351253177324030048, −2.22123095222546016500475416985, −1.26633319469863157650380625531, −0.927447940921103229909302939442, 0.927447940921103229909302939442, 1.26633319469863157650380625531, 2.22123095222546016500475416985, 2.44196075027351253177324030048, 3.34200650166558958121524680272, 3.45515698969463155512577447060, 4.22837792274623027778098214083, 4.24702578211883245021715298179, 4.89070470802441293896587460836, 5.11327606168201901615325305948, 5.72011169734652380728607285202, 5.84078569602366248045978915245, 6.49798999140538658481477969393, 6.88418751275801213484324672289, 7.50695737289222644858790861630, 7.53537913486923803595366408431, 8.039672198377512682103634579694, 8.190529888951037692011295760577, 8.869494181048331324556504621787, 9.037735255667995203201388236497

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