Properties

Label 4-1216e2-1.1-c1e2-0-1
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 3·9-s − 8·11-s − 13-s − 15-s − 3·17-s − 8·19-s + 5·23-s + 5·25-s + 8·27-s + 7·29-s − 8·31-s − 8·33-s − 20·37-s − 39-s + 5·41-s + 5·43-s − 3·45-s − 7·47-s − 14·49-s − 3·51-s + 11·53-s + 8·55-s − 8·57-s − 3·59-s + 11·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 9-s − 2.41·11-s − 0.277·13-s − 0.258·15-s − 0.727·17-s − 1.83·19-s + 1.04·23-s + 25-s + 1.53·27-s + 1.29·29-s − 1.43·31-s − 1.39·33-s − 3.28·37-s − 0.160·39-s + 0.780·41-s + 0.762·43-s − 0.447·45-s − 1.02·47-s − 2·49-s − 0.420·51-s + 1.51·53-s + 1.07·55-s − 1.05·57-s − 0.390·59-s + 1.40·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.165091905\)
\(L(\frac12)\) \(\approx\) \(1.165091905\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 11 T + 50 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−10.30635681697216334422864240023, −9.513383454135585988270874709104, −8.857571436506208770288316013988, −8.690712882522587051949739378897, −8.325139298921436122740996078547, −8.108139705660945225610688135763, −7.30575442479180154324138690813, −7.04099428933538834886565069078, −7.00387469099120421651974817908, −6.26606043798695773688729472289, −5.67637069027213819610780671363, −4.93970543655673286845490453258, −4.86486881921466456229017161779, −4.53854471935038728507819800624, −3.69116018372694259187518944766, −3.28546026377520809595273889941, −2.66361088398248151877003270588, −2.29035348611369716291379545667, −1.64565947226456425102207556606, −0.42643685515750211111448020377, 0.42643685515750211111448020377, 1.64565947226456425102207556606, 2.29035348611369716291379545667, 2.66361088398248151877003270588, 3.28546026377520809595273889941, 3.69116018372694259187518944766, 4.53854471935038728507819800624, 4.86486881921466456229017161779, 4.93970543655673286845490453258, 5.67637069027213819610780671363, 6.26606043798695773688729472289, 7.00387469099120421651974817908, 7.04099428933538834886565069078, 7.30575442479180154324138690813, 8.108139705660945225610688135763, 8.325139298921436122740996078547, 8.690712882522587051949739378897, 8.857571436506208770288316013988, 9.513383454135585988270874709104, 10.30635681697216334422864240023

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