Properties

Label 4-3840e2-1.1-c1e2-0-6
Degree $4$
Conductor $14745600$
Sign $1$
Analytic cond. $940.192$
Root an. cond. $5.53737$
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  − 9-s + 4·17-s − 25-s − 20·41-s − 16·47-s − 14·49-s − 16·71-s − 20·73-s + 81-s + 12·89-s + 4·97-s − 32·103-s + 4·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1/3·9-s + 0.970·17-s − 1/5·25-s − 3.12·41-s − 2.33·47-s − 2·49-s − 1.89·71-s − 2.34·73-s + 1/9·81-s + 1.27·89-s + 0.406·97-s − 3.15·103-s + 0.376·113-s + 6/11·121-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 − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(940.192\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6369973079\)
\(L(\frac12)\) \(\approx\) \(0.6369973079\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.635568669000205997865447065487, −8.321054569842023359020665420379, −7.939099336993195826249100835102, −7.70937619149048557652213230536, −7.14244581524667954109325995718, −6.80408937197220044945956823396, −6.55823237378054144170496292103, −5.94270219899862247246571898096, −5.85809434502423457454328825604, −5.10064812593398518040607489031, −5.07863490452185573928430053127, −4.56858212499066697266971008021, −4.11320977457367025366339236986, −3.36258123886322483064979221256, −3.29303287193962016839352071227, −2.96841844202890724872308200851, −2.17609484352965320585273225398, −1.51346847797634494888823595439, −1.44180050196798747769109932427, −0.22695768056695850267662283784, 0.22695768056695850267662283784, 1.44180050196798747769109932427, 1.51346847797634494888823595439, 2.17609484352965320585273225398, 2.96841844202890724872308200851, 3.29303287193962016839352071227, 3.36258123886322483064979221256, 4.11320977457367025366339236986, 4.56858212499066697266971008021, 5.07863490452185573928430053127, 5.10064812593398518040607489031, 5.85809434502423457454328825604, 5.94270219899862247246571898096, 6.55823237378054144170496292103, 6.80408937197220044945956823396, 7.14244581524667954109325995718, 7.70937619149048557652213230536, 7.939099336993195826249100835102, 8.321054569842023359020665420379, 8.635568669000205997865447065487

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