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Properties

Label 4-171e2-1.1-c1e2-0-1
Degree $4$
Conductor $29241$
Sign $1$
Analytic cond. $1.86443$
Root an. cond. $1.16852$
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 − 4-s + 5-s − 3·7-s + 8·8-s − 3·9-s − 2·10-s − 3·11-s − 12·13-s + 6·14-s − 7·16-s − 3·17-s + 6·18-s − 8·19-s − 20-s + 6·22-s + 16·23-s + 5·25-s + 24·26-s + 3·28-s + 5·29-s + 7·31-s − 14·32-s + 6·34-s − 3·35-s + 3·36-s + 4·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 0.447·5-s − 1.13·7-s + 2.82·8-s − 9-s − 0.632·10-s − 0.904·11-s − 3.32·13-s + 1.60·14-s − 7/4·16-s − 0.727·17-s + 1.41·18-s − 1.83·19-s − 0.223·20-s + 1.27·22-s + 3.33·23-s + 25-s + 4.70·26-s + 0.566·28-s + 0.928·29-s + 1.25·31-s − 2.47·32-s + 1.02·34-s − 0.507·35-s + 1/2·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29241\)    =    \(3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1.86443\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29241,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1815648274\)
\(L(\frac12)\) \(\approx\) \(0.1815648274\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.2.c_f
5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.23.aq_eg
29$C_2^2$ \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.29.af_ae
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} \) 2.41.ab_abo
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.47.j_bi
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2^2$ \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.71.ap_fy
73$C_2^2$ \( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.73.af_abw
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.79.y_lq
83$C_2^2$ \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \) 2.83.ab_ade
89$C_2^2$ \( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} \) 2.89.ab_adk
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−12.95945086820734451025233532554, −12.65628283020224663286541981072, −12.27823169827535240646540400128, −11.08727310085741671341719099340, −10.78828091209839798014757536336, −10.32300421738229913789095234517, −9.867798478870702478720766598629, −9.258254331395318973656095058231, −9.250357469858807382098256413935, −8.584132236487720075659785309418, −8.118392255684939794679506502188, −7.28610940937071882963108996979, −7.10729080348009551345534135960, −6.29817124545901915841628164405, −5.23762389817640505604613050252, −4.69639043709400729581956415633, −4.60496027176388281460416851243, −2.70785898672710861791754186834, −2.67118958128024930507435557314, −0.48960104281960340889231368470, 0.48960104281960340889231368470, 2.67118958128024930507435557314, 2.70785898672710861791754186834, 4.60496027176388281460416851243, 4.69639043709400729581956415633, 5.23762389817640505604613050252, 6.29817124545901915841628164405, 7.10729080348009551345534135960, 7.28610940937071882963108996979, 8.118392255684939794679506502188, 8.584132236487720075659785309418, 9.250357469858807382098256413935, 9.258254331395318973656095058231, 9.867798478870702478720766598629, 10.32300421738229913789095234517, 10.78828091209839798014757536336, 11.08727310085741671341719099340, 12.27823169827535240646540400128, 12.65628283020224663286541981072, 12.95945086820734451025233532554

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