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Properties

Label 4-33e4-1.1-c1e2-0-9
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $75.6153$
Root an. cond. $2.94884$
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  − 4·4-s + 12·16-s − 10·25-s + 22·31-s + 22·37-s + 11·49-s − 32·64-s + 22·67-s + 10·97-s + 40·100-s − 14·103-s − 88·124-s + 127-s + 131-s + 137-s + 139-s − 88·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 2·25-s + 3.95·31-s + 3.61·37-s + 11/7·49-s − 4·64-s + 2.68·67-s + 1.01·97-s + 4·100-s − 1.37·103-s − 7.90·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7.23·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(75.6153\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.318129229\)
\(L(\frac12)\) \(\approx\) \(1.318129229\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.31.aw_hb
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.37.aw_hn
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.a_er
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.a_afn
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.a_al
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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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

−8.157799715862211353426737114370, −7.85350411050304424040374786608, −7.40888065910781063604227830543, −6.40158080143163907333611081505, −6.34989393601049354736426581844, −5.65272361057678955740931295147, −5.44366242309248335244474381902, −4.59633679987356442004935711420, −4.46927736296465408535190629373, −4.10163309596027718433915629506, −3.57224945195733786318502793376, −2.81030152638481090628965604169, −2.33436956052754417074709546789, −1.04628882090551819052986550472, −0.68442490173741656332230083278, 0.68442490173741656332230083278, 1.04628882090551819052986550472, 2.33436956052754417074709546789, 2.81030152638481090628965604169, 3.57224945195733786318502793376, 4.10163309596027718433915629506, 4.46927736296465408535190629373, 4.59633679987356442004935711420, 5.44366242309248335244474381902, 5.65272361057678955740931295147, 6.34989393601049354736426581844, 6.40158080143163907333611081505, 7.40888065910781063604227830543, 7.85350411050304424040374786608, 8.157799715862211353426737114370

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