Properties

Label 4-901404-1.1-c1e2-0-5
Degree $4$
Conductor $901404$
Sign $1$
Analytic cond. $57.4743$
Root an. cond. $2.75339$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 4-s − 7-s + 6·9-s + 3·12-s + 16-s + 4·19-s − 3·21-s + 7·25-s + 9·27-s − 28-s + 2·31-s + 6·36-s − 2·37-s + 3·48-s + 49-s + 12·57-s − 13·61-s − 6·63-s + 64-s + 21·67-s − 4·73-s + 21·75-s + 4·76-s − 10·79-s + 9·81-s − 3·84-s + ⋯
L(s)  = 1  + 1.73·3-s + 1/2·4-s − 0.377·7-s + 2·9-s + 0.866·12-s + 1/4·16-s + 0.917·19-s − 0.654·21-s + 7/5·25-s + 1.73·27-s − 0.188·28-s + 0.359·31-s + 36-s − 0.328·37-s + 0.433·48-s + 1/7·49-s + 1.58·57-s − 1.66·61-s − 0.755·63-s + 1/8·64-s + 2.56·67-s − 0.468·73-s + 2.42·75-s + 0.458·76-s − 1.12·79-s + 81-s − 0.327·84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(901404\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 73\)
Sign: $1$
Analytic conductor: \(57.4743\)
Root analytic conductor: \(2.75339\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{901404} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 901404,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.899444515\)
\(L(\frac12)\) \(\approx\) \(4.899444515\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 - p T + p T^{2} \)
7$C_1$ \( 1 + T \)
73$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 5 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \)
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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

−8.212940248075370897129165098077, −7.80348401917433760724990858699, −7.29623744391917286086061549651, −7.04559011370954116880459693422, −6.59557016596219740830159217958, −6.07854688706901528173685618462, −5.43212099226806004781363222020, −4.90765128538370930874620909369, −4.36535165530639452638801602593, −3.77487566333540990485729092329, −3.23435846117427372584287979412, −2.95305606980784467193837350316, −2.40973903976661527989271613546, −1.73692949903066074477967300274, −0.998831840589346850229603562541, 0.998831840589346850229603562541, 1.73692949903066074477967300274, 2.40973903976661527989271613546, 2.95305606980784467193837350316, 3.23435846117427372584287979412, 3.77487566333540990485729092329, 4.36535165530639452638801602593, 4.90765128538370930874620909369, 5.43212099226806004781363222020, 6.07854688706901528173685618462, 6.59557016596219740830159217958, 7.04559011370954116880459693422, 7.29623744391917286086061549651, 7.80348401917433760724990858699, 8.212940248075370897129165098077

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