Properties

Label 4-429e2-1.1-c1e2-0-10
Degree $4$
Conductor $184041$
Sign $1$
Analytic cond. $11.7346$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 4-s − 4·5-s − 4·6-s − 4·7-s + 3·9-s + 8·10-s + 2·11-s + 2·12-s − 2·13-s + 8·14-s − 8·15-s + 16-s − 8·17-s − 6·18-s − 12·19-s − 4·20-s − 8·21-s − 4·22-s + 4·23-s + 4·25-s + 4·26-s + 4·27-s − 4·28-s + 16·30-s + 2·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.78·5-s − 1.63·6-s − 1.51·7-s + 9-s + 2.52·10-s + 0.603·11-s + 0.577·12-s − 0.554·13-s + 2.13·14-s − 2.06·15-s + 1/4·16-s − 1.94·17-s − 1.41·18-s − 2.75·19-s − 0.894·20-s − 1.74·21-s − 0.852·22-s + 0.834·23-s + 4/5·25-s + 0.784·26-s + 0.769·27-s − 0.755·28-s + 2.92·30-s + 0.353·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(184041\)    =    \(3^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(11.7346\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 184041,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_d
5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_m
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_k
17$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bw
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bq
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.31.a_bs
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_bu
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.41.y_is
43$D_{4}$ \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_bo
47$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.47.a_w
53$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_as
59$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_gs
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$D_{4}$ \( 1 - 8 T + 100 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_dw
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.71.a_eg
73$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_dm
79$D_{4}$ \( 1 + 4 T + 112 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_ei
83$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.83.a_bm
89$D_{4}$ \( 1 + 28 T + 372 T^{2} + 28 p T^{3} + p^{2} T^{4} \) 2.89.bc_oi
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−10.73462947775384612173659269797, −10.40713923653342989842260535016, −9.683909530252162471608031576754, −9.572267351407206440797229255036, −8.977965731951957614047021142668, −8.625211138176998733528565617552, −8.294380618832128264164396604935, −8.198025685798010879257008486586, −7.31886910412356544374495004208, −6.77392418873735157134334304512, −6.73456846493927341009354436701, −6.11499320682043626504615835088, −4.63610094777111649562187410115, −4.50784304739186640667264156397, −3.77219787946851793509270493000, −3.39557814695746474405673163071, −2.65702480055049271571845253411, −1.85487998435817000029845974424, 0, 0, 1.85487998435817000029845974424, 2.65702480055049271571845253411, 3.39557814695746474405673163071, 3.77219787946851793509270493000, 4.50784304739186640667264156397, 4.63610094777111649562187410115, 6.11499320682043626504615835088, 6.73456846493927341009354436701, 6.77392418873735157134334304512, 7.31886910412356544374495004208, 8.198025685798010879257008486586, 8.294380618832128264164396604935, 8.625211138176998733528565617552, 8.977965731951957614047021142668, 9.572267351407206440797229255036, 9.683909530252162471608031576754, 10.40713923653342989842260535016, 10.73462947775384612173659269797

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