Properties

Label 4-771228-1.1-c1e2-0-2
Degree $4$
Conductor $771228$
Sign $1$
Analytic cond. $49.1741$
Root an. cond. $2.64809$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s − 2·7-s + 9-s + 12-s − 6·13-s + 16-s − 8·19-s + 2·21-s − 4·25-s − 27-s + 2·28-s − 14·31-s − 36-s + 3·37-s + 6·39-s − 10·43-s − 48-s − 10·49-s + 6·52-s + 8·57-s + 4·61-s − 2·63-s − 64-s − 12·67-s − 10·73-s + 4·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 1/4·16-s − 1.83·19-s + 0.436·21-s − 4/5·25-s − 0.192·27-s + 0.377·28-s − 2.51·31-s − 1/6·36-s + 0.493·37-s + 0.960·39-s − 1.52·43-s − 0.144·48-s − 1.42·49-s + 0.832·52-s + 1.05·57-s + 0.512·61-s − 0.251·63-s − 1/8·64-s − 1.46·67-s − 1.17·73-s + 0.461·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 771228 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 771228 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(771228\)    =    \(2^{2} \cdot 3^{3} \cdot 37 \cdot 193\)
Sign: $1$
Analytic conductor: \(49.1741\)
Root analytic conductor: \(2.64809\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{771228} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 771228,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 + T \)
37$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
193$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 152 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−7.79224455657778699383124354578, −7.29292436158084576345184823278, −6.95352607934367056218455621176, −6.42863035110409498933366257922, −6.07270193358115412142400173279, −5.42252687682940923322983945422, −5.18976692514709859495414520507, −4.44014413390248417105686697959, −4.25228170186027373727845078814, −3.53130900212502161971790500664, −2.99197885457657002070145192126, −2.16367848633616435420462973109, −1.67716142500622267233069906139, 0, 0, 1.67716142500622267233069906139, 2.16367848633616435420462973109, 2.99197885457657002070145192126, 3.53130900212502161971790500664, 4.25228170186027373727845078814, 4.44014413390248417105686697959, 5.18976692514709859495414520507, 5.42252687682940923322983945422, 6.07270193358115412142400173279, 6.42863035110409498933366257922, 6.95352607934367056218455621176, 7.29292436158084576345184823278, 7.79224455657778699383124354578

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