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Properties

Label 4-37971-1.1-c1e2-0-0
Degree $4$
Conductor $37971$
Sign $1$
Analytic cond. $2.42106$
Root an. cond. $1.24738$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·4-s − 7·7-s + 9-s + 6·12-s − 10·13-s + 5·16-s + 2·19-s + 14·21-s − 7·25-s + 4·27-s + 21·28-s − 5·31-s − 3·36-s − 6·37-s + 20·39-s − 7·43-s − 10·48-s + 25·49-s + 30·52-s − 4·57-s + 17·61-s − 7·63-s − 3·64-s − 4·67-s − 11·73-s + 14·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s − 2.64·7-s + 1/3·9-s + 1.73·12-s − 2.77·13-s + 5/4·16-s + 0.458·19-s + 3.05·21-s − 7/5·25-s + 0.769·27-s + 3.96·28-s − 0.898·31-s − 1/2·36-s − 0.986·37-s + 3.20·39-s − 1.06·43-s − 1.44·48-s + 25/7·49-s + 4.16·52-s − 0.529·57-s + 2.17·61-s − 0.881·63-s − 3/8·64-s − 0.488·67-s − 1.28·73-s + 1.61·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(37971\)    =    \(3^{2} \cdot 4219\)
Sign: $1$
Analytic conductor: \(2.42106\)
Root analytic conductor: \(1.24738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 37971,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + 2 T + p T^{2} \)
4219$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 124 T + p T^{2} ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.h_y
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.k_by
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.19.ac_bm
23$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.23.a_abc
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.f_ba
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.41.a_f
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.h_da
47$C_2^2$ \( 1 - 81 T^{2} + p^{2} T^{4} \) 2.47.a_add
53$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.53.a_adc
59$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.59.a_u
61$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.ar_fw
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.e_ag
71$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \) 2.71.a_aen
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.l_gs
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.79.h_gc
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \) 2.89.a_dr
97$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.97.a_adr
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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

−9.730356080882589254251777600795, −9.614151100927429702240158621768, −9.037985727910095804199142600062, −8.354203302416302534494413578631, −7.43936086076760350291685890369, −7.02864048579277621030611895916, −6.54578564582145848168097550746, −5.75291311764833026998670840473, −5.37639358692116158053118938575, −4.86189612245815922762630911511, −4.05124127270206260791712491337, −3.38807780941466449294633521090, −2.58225597439718439035264744581, 0, 0, 2.58225597439718439035264744581, 3.38807780941466449294633521090, 4.05124127270206260791712491337, 4.86189612245815922762630911511, 5.37639358692116158053118938575, 5.75291311764833026998670840473, 6.54578564582145848168097550746, 7.02864048579277621030611895916, 7.43936086076760350291685890369, 8.354203302416302534494413578631, 9.037985727910095804199142600062, 9.614151100927429702240158621768, 9.730356080882589254251777600795

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