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Properties

Label 4-1053e2-1.1-c1e2-0-27
Degree $4$
Conductor $1108809$
Sign $1$
Analytic cond. $70.6986$
Root an. cond. $2.89969$
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-s − 2·4-s − 3·5-s − 3·7-s − 3·8-s − 3·10-s + 5·11-s − 2·13-s − 3·14-s + 16-s + 4·17-s − 6·19-s + 6·20-s + 5·22-s − 5·23-s − 2·25-s − 2·26-s + 6·28-s − 5·29-s + 31-s + 2·32-s + 4·34-s + 9·35-s − 37-s − 6·38-s + 9·40-s − 11·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s − 1.34·5-s − 1.13·7-s − 1.06·8-s − 0.948·10-s + 1.50·11-s − 0.554·13-s − 0.801·14-s + 1/4·16-s + 0.970·17-s − 1.37·19-s + 1.34·20-s + 1.06·22-s − 1.04·23-s − 2/5·25-s − 0.392·26-s + 1.13·28-s − 0.928·29-s + 0.179·31-s + 0.353·32-s + 0.685·34-s + 1.52·35-s − 0.164·37-s − 0.973·38-s + 1.42·40-s − 1.71·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1108809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1108809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1108809\)    =    \(3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(70.6986\)
Root analytic conductor: \(2.89969\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1108809,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
5$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_l
7$D_{4}$ \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_p
11$D_{4}$ \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_r
17$D_{4}$ \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_bh
19$D_{4}$ \( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bb
23$D_{4}$ \( 1 + 5 T + 41 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.23.f_bp
29$D_{4}$ \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.29.f_cb
31$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_b
37$D_{4}$ \( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} \) 2.37.b_br
41$C_4$ \( 1 + 11 T + 111 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.41.l_eh
43$D_{4}$ \( 1 + 16 T + 145 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.43.q_fp
47$D_{4}$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_df
53$D_{4}$ \( 1 - 14 T + 110 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.53.ao_eg
59$D_{4}$ \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_dv
61$D_{4}$ \( 1 + 5 T + 97 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_dt
67$D_{4}$ \( 1 + 7 T + 115 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_el
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.71.ak_gl
73$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.73.a_v
79$C_2^2$ \( 1 + 153 T^{2} + p^{2} T^{4} \) 2.79.a_fx
83$D_{4}$ \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.83.ai_cf
89$D_{4}$ \( 1 + 3 T + 179 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.89.d_gx
97$D_{4}$ \( 1 + 4 T + 178 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_gw
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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

−9.629505132742867985506190112397, −9.402358213255762819935690877782, −8.722203097877484885906989865646, −8.606729185751585374711267327116, −8.078314143771985418444226321739, −7.67745771191599407757193972290, −7.12232866859390497374116521227, −6.69161541022450792844172287419, −6.26297154661453989759920105921, −5.90902952612023712179843526101, −5.17206257302330982387273116887, −4.89851888944472995285731917910, −4.07120430693461348223116487038, −4.04445557174768138416560211291, −3.47322911231444295352018567693, −3.43002708804291179789827397212, −2.31435365180976146227990411929, −1.47145655271924622801729542746, 0, 0, 1.47145655271924622801729542746, 2.31435365180976146227990411929, 3.43002708804291179789827397212, 3.47322911231444295352018567693, 4.04445557174768138416560211291, 4.07120430693461348223116487038, 4.89851888944472995285731917910, 5.17206257302330982387273116887, 5.90902952612023712179843526101, 6.26297154661453989759920105921, 6.69161541022450792844172287419, 7.12232866859390497374116521227, 7.67745771191599407757193972290, 8.078314143771985418444226321739, 8.606729185751585374711267327116, 8.722203097877484885906989865646, 9.402358213255762819935690877782, 9.629505132742867985506190112397

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