Properties

Label 4-189e2-1.1-c1e2-0-1
Degree $4$
Conductor $35721$
Sign $1$
Analytic cond. $2.27760$
Root an. cond. $1.22848$
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  − 2·4-s + 7-s + 9·19-s + 5·25-s − 2·28-s − 3·31-s − 10·37-s + 26·43-s − 6·49-s − 15·61-s + 8·64-s + 16·67-s + 27·73-s − 18·76-s + 4·79-s − 10·100-s + 6·103-s − 19·109-s − 11·121-s + 6·124-s + ⋯
L(s)  = 1  − 4-s + 0.377·7-s + 2.06·19-s + 25-s − 0.377·28-s − 0.538·31-s − 1.64·37-s + 3.96·43-s − 6/7·49-s − 1.92·61-s + 64-s + 1.95·67-s + 3.16·73-s − 2.06·76-s + 0.450·79-s − 100-s + 0.591·103-s − 1.81·109-s − 121-s + 0.538·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(35721\)    =    \(3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2.27760\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 35721,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.068089793\)
\(L(\frac12)\) \(\approx\) \(1.068089793\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.aj_bu
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.d_bi
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.k_cl
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.43.aba_jv
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.53.a_cb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.p_fg
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.67.aq_hh
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.73.abb_me
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.a_gn
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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

−12.55212682660438983472157645227, −12.54713968965605036693842573810, −11.97555252763756146134648577293, −11.08995198316239262165038813214, −11.05401100186968864581132446171, −10.38152969226145926335818366762, −9.602976770644561260497905342236, −9.288465943745865816007739079544, −9.036360803211178457930941565742, −8.270771669586352006171892199710, −7.76823041981533334697404465071, −7.27141302921257274556978944484, −6.65677308889071393553363619996, −5.80736724567364436227539884582, −5.14979491245933178669563729918, −4.91529950925449047407897129273, −4.00740754937879545210542399250, −3.42704674854982646766031179707, −2.43514023894947911090584523261, −1.03997947285202387552651717274, 1.03997947285202387552651717274, 2.43514023894947911090584523261, 3.42704674854982646766031179707, 4.00740754937879545210542399250, 4.91529950925449047407897129273, 5.14979491245933178669563729918, 5.80736724567364436227539884582, 6.65677308889071393553363619996, 7.27141302921257274556978944484, 7.76823041981533334697404465071, 8.270771669586352006171892199710, 9.036360803211178457930941565742, 9.288465943745865816007739079544, 9.602976770644561260497905342236, 10.38152969226145926335818366762, 11.05401100186968864581132446171, 11.08995198316239262165038813214, 11.97555252763756146134648577293, 12.54713968965605036693842573810, 12.55212682660438983472157645227

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