Properties

Label 4-1216e2-1.1-c1e2-0-16
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·5-s + 3·9-s + 8·11-s − 5·13-s + 3·15-s + 5·17-s − 8·19-s + 23-s + 5·25-s + 8·27-s + 3·29-s + 8·31-s + 8·33-s − 4·37-s − 5·39-s + 5·41-s − 11·43-s + 9·45-s + 5·47-s − 14·49-s + 5·51-s − 9·53-s + 24·55-s − 8·57-s + 13·59-s − 61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 9-s + 2.41·11-s − 1.38·13-s + 0.774·15-s + 1.21·17-s − 1.83·19-s + 0.208·23-s + 25-s + 1.53·27-s + 0.557·29-s + 1.43·31-s + 1.39·33-s − 0.657·37-s − 0.800·39-s + 0.780·41-s − 1.67·43-s + 1.34·45-s + 0.729·47-s − 2·49-s + 0.700·51-s − 1.23·53-s + 3.23·55-s − 1.05·57-s + 1.69·59-s − 0.128·61-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

\(L(1)\) \(\approx\) \(4.680950561\)
\(L(\frac12)\) \(\approx\) \(4.680950561\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−10.09218622119851267971677131830, −9.623336740363681090136830860476, −9.185990198395943750064219674179, −8.618370361336913708281870799250, −8.577284145241727260502725660451, −8.047099642116034488546395432749, −7.16037949655193162309034997164, −7.13509691775498423330180611495, −6.59867913544136811142849482496, −6.19763187865265040010956589756, −6.03593189324504591406776830332, −5.16188938104293955838932428878, −4.62288120007408061392660709754, −4.49860334420986502597813157110, −3.85058148625644840414696704793, −3.16848134761799190016259149630, −2.76958147167755201392029308328, −1.92866026723187867845631256551, −1.65343395986600288155362959938, −0.973127705026115125339362395937, 0.973127705026115125339362395937, 1.65343395986600288155362959938, 1.92866026723187867845631256551, 2.76958147167755201392029308328, 3.16848134761799190016259149630, 3.85058148625644840414696704793, 4.49860334420986502597813157110, 4.62288120007408061392660709754, 5.16188938104293955838932428878, 6.03593189324504591406776830332, 6.19763187865265040010956589756, 6.59867913544136811142849482496, 7.13509691775498423330180611495, 7.16037949655193162309034997164, 8.047099642116034488546395432749, 8.577284145241727260502725660451, 8.618370361336913708281870799250, 9.185990198395943750064219674179, 9.623336740363681090136830860476, 10.09218622119851267971677131830

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