Properties

Label 4-1216e2-1.1-c1e2-0-2
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  + 2·7-s − 3·9-s − 10·17-s − 6·23-s − 6·25-s + 20·31-s − 12·41-s + 16·47-s − 11·49-s − 6·63-s − 24·71-s + 22·73-s + 32·79-s − 8·89-s − 24·97-s − 4·103-s + 16·113-s − 20·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s + ⋯
L(s)  = 1  + 0.755·7-s − 9-s − 2.42·17-s − 1.25·23-s − 6/5·25-s + 3.59·31-s − 1.87·41-s + 2.33·47-s − 1.57·49-s − 0.755·63-s − 2.84·71-s + 2.57·73-s + 3.60·79-s − 0.847·89-s − 2.43·97-s − 0.394·103-s + 1.50·113-s − 1.83·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.42·153-s + 0.0798·157-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

\(L(1)\) \(\approx\) \(1.213666843\)
\(L(\frac12)\) \(\approx\) \(1.213666843\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−10.34162321963084756228495507212, −9.789944851281245100404629534036, −8.888684763342553795967179645308, −8.702666780100205789920154436905, −8.349699421063129414052767235301, −7.931113381001288443833542932298, −7.74981057464855372059181133882, −6.82861611084092418158079627366, −6.66782394428390095259465866418, −6.09620023331127049825297063168, −5.97086886665497760988066145289, −5.06095169685955457986079072353, −4.92058352265273508167847449905, −4.23343298498192265577536023043, −4.07736104185211590370685136133, −3.23270879701863519368115653074, −2.55192185855878262092093367963, −2.25676566650142583540662821214, −1.59768397922571576498652599915, −0.45615319848583992520295124011, 0.45615319848583992520295124011, 1.59768397922571576498652599915, 2.25676566650142583540662821214, 2.55192185855878262092093367963, 3.23270879701863519368115653074, 4.07736104185211590370685136133, 4.23343298498192265577536023043, 4.92058352265273508167847449905, 5.06095169685955457986079072353, 5.97086886665497760988066145289, 6.09620023331127049825297063168, 6.66782394428390095259465866418, 6.82861611084092418158079627366, 7.74981057464855372059181133882, 7.931113381001288443833542932298, 8.349699421063129414052767235301, 8.702666780100205789920154436905, 8.888684763342553795967179645308, 9.789944851281245100404629534036, 10.34162321963084756228495507212

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