Properties

Label 4-3593700-1.1-c1e2-0-6
Degree $4$
Conductor $3593700$
Sign $1$
Analytic cond. $229.137$
Root an. cond. $3.89066$
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-s + 4-s − 5-s − 2·9-s − 11-s + 12-s − 15-s + 16-s − 20-s + 9·23-s − 4·25-s − 5·27-s + 5·31-s − 33-s − 2·36-s + 3·37-s − 44-s + 2·45-s − 14·47-s + 48-s − 4·49-s − 53-s + 55-s + 2·59-s − 60-s + 64-s + 21·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 0.447·5-s − 2/3·9-s − 0.301·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.223·20-s + 1.87·23-s − 4/5·25-s − 0.962·27-s + 0.898·31-s − 0.174·33-s − 1/3·36-s + 0.493·37-s − 0.150·44-s + 0.298·45-s − 2.04·47-s + 0.144·48-s − 4/7·49-s − 0.137·53-s + 0.134·55-s + 0.260·59-s − 0.129·60-s + 1/8·64-s + 2.56·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3593700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3593700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3593700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(229.137\)
Root analytic conductor: \(3.89066\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3593700,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.274852668\)
\(L(\frac12)\) \(\approx\) \(2.274852668\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
5$C_2$ \( 1 + T + p T^{2} \)
11$C_1$ \( 1 + T \)
good7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \)
41$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 100 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 145 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
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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

−7.54798556187355784128127009202, −7.04958608428211742170617853400, −6.72555239054623468957302521595, −6.31402394403699078311772463822, −5.83830544905062922637504815692, −5.37069723012696154121308522527, −4.93684821270376645163559275615, −4.56657789180832519087621453248, −3.87975242055993376574286856629, −3.45543102252062312787811173932, −3.00497682923362451750411401934, −2.67076226600141549066376266475, −2.07254581752859807525680301552, −1.40568876836640456695701174059, −0.53212271449243159413008556043, 0.53212271449243159413008556043, 1.40568876836640456695701174059, 2.07254581752859807525680301552, 2.67076226600141549066376266475, 3.00497682923362451750411401934, 3.45543102252062312787811173932, 3.87975242055993376574286856629, 4.56657789180832519087621453248, 4.93684821270376645163559275615, 5.37069723012696154121308522527, 5.83830544905062922637504815692, 6.31402394403699078311772463822, 6.72555239054623468957302521595, 7.04958608428211742170617853400, 7.54798556187355784128127009202

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