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Properties

Label 4-334e2-1.1-c1e2-0-0
Degree $4$
Conductor $111556$
Sign $1$
Analytic cond. $7.11290$
Root an. cond. $1.63309$
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  − 2·2-s + 3·4-s + 2·5-s − 6·7-s − 4·8-s + 2·9-s − 4·10-s + 8·13-s + 12·14-s + 5·16-s + 4·17-s − 4·18-s + 4·19-s + 6·20-s + 12·23-s − 5·25-s − 16·26-s − 18·28-s + 8·29-s − 10·31-s − 6·32-s − 8·34-s − 12·35-s + 6·36-s − 6·37-s − 8·38-s − 8·40-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.894·5-s − 2.26·7-s − 1.41·8-s + 2/3·9-s − 1.26·10-s + 2.21·13-s + 3.20·14-s + 5/4·16-s + 0.970·17-s − 0.942·18-s + 0.917·19-s + 1.34·20-s + 2.50·23-s − 25-s − 3.13·26-s − 3.40·28-s + 1.48·29-s − 1.79·31-s − 1.06·32-s − 1.37·34-s − 2.02·35-s + 36-s − 0.986·37-s − 1.29·38-s − 1.26·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(111556\)    =    \(2^{2} \cdot 167^{2}\)
Sign: $1$
Analytic conductor: \(7.11290\)
Root analytic conductor: \(1.63309\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 111556,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9137859341\)
\(L(\frac12)\) \(\approx\) \(0.9137859341\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
167$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_j
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bi
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_be
19$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_k
23$C_4$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.23.am_cw
29$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_co
31$D_{4}$ \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_db
37$D_{4}$ \( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_cn
41$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_di
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.43.ae_dm
47$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_bf
53$D_{4}$ \( 1 + 22 T + 225 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.53.w_ir
59$D_{4}$ \( 1 - 10 T + 141 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_fl
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.61.a_by
67$D_{4}$ \( 1 + 2 T + 133 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_fd
71$D_{4}$ \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_ec
73$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_gw
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.79.a_di
83$D_{4}$ \( 1 - 10 T + 93 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_dp
89$D_{4}$ \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_fz
97$D_{4}$ \( 1 - 2 T + 187 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_hf
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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

−11.40862124668015476417861597307, −11.25831855571234334024248018304, −10.63836262376064049293827407208, −10.27675099742077063652552279128, −9.797943074436980854110866835969, −9.427982635605966008212835267409, −9.220763079702653755822206274757, −8.804495338373476011558112604988, −8.105460500173576049875070469091, −7.48843041404042231470088965312, −7.01382435850729352939952586464, −6.54972548942487316992370508552, −6.07967418279830285609275530825, −5.85110692940051632916534747545, −5.03775345127947808510550751681, −3.72695237079035098531648530366, −3.30740996308260878574767635524, −2.90683106087226640508775420160, −1.62732020919446501342281036349, −0.936687410837248384545522995092, 0.936687410837248384545522995092, 1.62732020919446501342281036349, 2.90683106087226640508775420160, 3.30740996308260878574767635524, 3.72695237079035098531648530366, 5.03775345127947808510550751681, 5.85110692940051632916534747545, 6.07967418279830285609275530825, 6.54972548942487316992370508552, 7.01382435850729352939952586464, 7.48843041404042231470088965312, 8.105460500173576049875070469091, 8.804495338373476011558112604988, 9.220763079702653755822206274757, 9.427982635605966008212835267409, 9.797943074436980854110866835969, 10.27675099742077063652552279128, 10.63836262376064049293827407208, 11.25831855571234334024248018304, 11.40862124668015476417861597307

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