Properties

Label 4-189e2-1.1-c1e2-0-12
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  + 4·4-s + 7-s + 12·16-s − 10·25-s + 4·28-s − 22·37-s − 16·43-s − 6·49-s + 32·64-s + 10·67-s + 34·79-s − 40·100-s − 4·109-s + 12·112-s + 22·121-s + 127-s + 131-s + 137-s + 139-s − 88·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s − 64·172-s + ⋯
L(s)  = 1  + 2·4-s + 0.377·7-s + 3·16-s − 2·25-s + 0.755·28-s − 3.61·37-s − 2.43·43-s − 6/7·49-s + 4·64-s + 1.22·67-s + 3.82·79-s − 4·100-s − 0.383·109-s + 1.13·112-s + 2·121-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 − 0.0769·169-s − 4.87·172-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: induced by $\chi_{189} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 35721,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.043182272\)
\(L(\frac12)\) \(\approx\) \(2.043182272\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−12.34724376774533062031826258018, −12.31743923970937630724778633707, −11.77926194512005936178483948718, −11.44993374274485632181077985580, −10.76104049195539061702592615885, −10.66527287713730006787258232932, −9.806373410543754724936667305845, −9.715222368130451539338716214644, −8.404676935173241946472510450721, −8.338971354451228504286981770129, −7.60763725919989967678800386198, −7.08463020998111621518733980564, −6.66585914897251330746172488364, −6.14132080213471396691959727237, −5.43995460113139782364315789736, −4.96364125632316960128323566378, −3.53797810840314301150019977587, −3.41334002214837305559225227638, −2.05946601437150731894798750794, −1.79667931384199699777336608332, 1.79667931384199699777336608332, 2.05946601437150731894798750794, 3.41334002214837305559225227638, 3.53797810840314301150019977587, 4.96364125632316960128323566378, 5.43995460113139782364315789736, 6.14132080213471396691959727237, 6.66585914897251330746172488364, 7.08463020998111621518733980564, 7.60763725919989967678800386198, 8.338971354451228504286981770129, 8.404676935173241946472510450721, 9.715222368130451539338716214644, 9.806373410543754724936667305845, 10.66527287713730006787258232932, 10.76104049195539061702592615885, 11.44993374274485632181077985580, 11.77926194512005936178483948718, 12.31743923970937630724778633707, 12.34724376774533062031826258018

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