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Properties

Label 4-61191-1.1-c1e2-0-0
Degree $4$
Conductor $61191$
Sign $-1$
Analytic cond. $3.90159$
Root an. cond. $1.40543$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·4-s + 7-s − 2·9-s − 3·12-s − 4·13-s + 5·16-s + 19-s + 21-s + 2·25-s − 5·27-s − 3·28-s + 4·31-s + 6·36-s − 7·37-s − 4·39-s − 7·43-s + 5·48-s − 13·49-s + 12·52-s + 57-s − 13·61-s − 2·63-s − 3·64-s − 4·67-s − 14·73-s + 2·75-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s + 0.377·7-s − 2/3·9-s − 0.866·12-s − 1.10·13-s + 5/4·16-s + 0.229·19-s + 0.218·21-s + 2/5·25-s − 0.962·27-s − 0.566·28-s + 0.718·31-s + 36-s − 1.15·37-s − 0.640·39-s − 1.06·43-s + 0.721·48-s − 1.85·49-s + 1.66·52-s + 0.132·57-s − 1.66·61-s − 0.251·63-s − 3/8·64-s − 0.488·67-s − 1.63·73-s + 0.230·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61191 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61191 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(61191\)    =    \(3^{2} \cdot 13 \cdot 523\)
Sign: $-1$
Analytic conductor: \(3.90159\)
Root analytic conductor: \(1.40543\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 61191,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 5 T + p T^{2} ) \)
523$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 16 T + p T^{2} ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ab_o
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.17.a_ag
19$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.ab_ae
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.29.a_z
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ae_ck
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.h_be
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.h_du
47$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.47.a_v
53$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.53.a_dd
59$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.59.a_aw
61$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.n_fe
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.71.a_bo
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.79.u_jx
83$C_2^2$ \( 1 + 159 T^{2} + p^{2} T^{4} \) 2.83.a_gd
89$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.89.a_adu
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.b_fw
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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

−9.598166945675670809041781490934, −9.044684732914527748756004052245, −8.768860707139523857397320172608, −8.293127274655682560377660322928, −7.80504630443563405451670430567, −7.31860576555547480231858828276, −6.55267102607826179291317800028, −5.81139164465520070299020970922, −5.17822374389000015157038597716, −4.75640716598523457455490532851, −4.28210020548262894700252823611, −3.33267497138655256743156315826, −2.88924014116876872958249383166, −1.69943857900062206966245237642, 0, 1.69943857900062206966245237642, 2.88924014116876872958249383166, 3.33267497138655256743156315826, 4.28210020548262894700252823611, 4.75640716598523457455490532851, 5.17822374389000015157038597716, 5.81139164465520070299020970922, 6.55267102607826179291317800028, 7.31860576555547480231858828276, 7.80504630443563405451670430567, 8.293127274655682560377660322928, 8.768860707139523857397320172608, 9.044684732914527748756004052245, 9.598166945675670809041781490934

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