Properties

Label 4-471e2-1.1-c1e2-0-1
Degree $4$
Conductor $221841$
Sign $-1$
Analytic cond. $14.1447$
Root an. cond. $1.93931$
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  + 4-s + 9-s − 4·11-s + 2·13-s − 3·16-s − 2·17-s − 4·19-s − 2·25-s + 8·31-s + 36-s − 14·37-s − 4·44-s − 8·47-s − 49-s + 2·52-s − 7·64-s − 2·68-s − 12·71-s − 4·76-s + 81-s − 18·89-s − 4·99-s − 2·100-s + 10·101-s + 2·109-s − 2·113-s + 2·117-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 3/4·16-s − 0.485·17-s − 0.917·19-s − 2/5·25-s + 1.43·31-s + 1/6·36-s − 2.30·37-s − 0.603·44-s − 1.16·47-s − 1/7·49-s + 0.277·52-s − 7/8·64-s − 0.242·68-s − 1.42·71-s − 0.458·76-s + 1/9·81-s − 1.90·89-s − 0.402·99-s − 1/5·100-s + 0.995·101-s + 0.191·109-s − 0.188·113-s + 0.184·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(221841\)    =    \(3^{2} \cdot 157^{2}\)
Sign: $-1$
Analytic conductor: \(14.1447\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 221841,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
157$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 138 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \)
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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

−8.600880989554269961839054807695, −8.466334499617947271307519615289, −7.82518434363817597256867507777, −7.35475260706879974578016860262, −6.74482293091353902445130051099, −6.54988525104159795299261648224, −5.90507484735023464570841611575, −5.35474862672613164845552761341, −4.68761726172784495564637266647, −4.37084941620788759664953324954, −3.53457398354938014465723819771, −2.89319909526715845526126540440, −2.23534806636514928119903856201, −1.59131130085401414686637681654, 0, 1.59131130085401414686637681654, 2.23534806636514928119903856201, 2.89319909526715845526126540440, 3.53457398354938014465723819771, 4.37084941620788759664953324954, 4.68761726172784495564637266647, 5.35474862672613164845552761341, 5.90507484735023464570841611575, 6.54988525104159795299261648224, 6.74482293091353902445130051099, 7.35475260706879974578016860262, 7.82518434363817597256867507777, 8.466334499617947271307519615289, 8.600880989554269961839054807695

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