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Properties

Label 4-171e2-1.1-c1e2-0-6
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s + 2·4-s − 2·5-s + 3·6-s − 3·7-s + 5·8-s + 6·9-s − 2·10-s − 3·11-s + 6·12-s + 6·13-s − 3·14-s − 6·15-s + 5·16-s − 3·17-s + 6·18-s − 8·19-s − 4·20-s − 9·21-s − 3·22-s − 8·23-s + 15·24-s − 7·25-s + 6·26-s + 9·27-s − 6·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 4-s − 0.894·5-s + 1.22·6-s − 1.13·7-s + 1.76·8-s + 2·9-s − 0.632·10-s − 0.904·11-s + 1.73·12-s + 1.66·13-s − 0.801·14-s − 1.54·15-s + 5/4·16-s − 0.727·17-s + 1.41·18-s − 1.83·19-s − 0.894·20-s − 1.96·21-s − 0.639·22-s − 1.66·23-s + 3.06·24-s − 7/5·25-s + 1.17·26-s + 1.73·27-s − 1.13·28-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

\(L(1)\) \(\approx\) \(2.905037239\)
\(L(\frac12)\) \(\approx\) \(2.905037239\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_ab
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
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$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
23$C_2^2$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_bp
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.29.k_df
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$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.41.c_df
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.47.as_gt
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.59.ag_ex
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
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$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_cn
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$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−13.01135616862345496556186435003, −12.97521832623334953220265396399, −12.14026648587108730965363885108, −11.59094825265085010171728403008, −10.86324642580606885432002952431, −10.70703578334391991843956196728, −9.920853486335495210077965541604, −9.662682133372304131530054979206, −8.598884274685412542893589621993, −8.402380373340848535792101374326, −7.74510585684968636519446495978, −7.60034681006466857306184100899, −6.56471602709823055667951663543, −6.43824348244610995544157926519, −5.48619817971487371812528013787, −4.20616057459367871776702764288, −3.86507867123677708209267961929, −3.72366236420306153594793405401, −2.38020460520154617673660805410, −2.11777052431038888001191696307, 2.11777052431038888001191696307, 2.38020460520154617673660805410, 3.72366236420306153594793405401, 3.86507867123677708209267961929, 4.20616057459367871776702764288, 5.48619817971487371812528013787, 6.43824348244610995544157926519, 6.56471602709823055667951663543, 7.60034681006466857306184100899, 7.74510585684968636519446495978, 8.402380373340848535792101374326, 8.598884274685412542893589621993, 9.662682133372304131530054979206, 9.920853486335495210077965541604, 10.70703578334391991843956196728, 10.86324642580606885432002952431, 11.59094825265085010171728403008, 12.14026648587108730965363885108, 12.97521832623334953220265396399, 13.01135616862345496556186435003

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