Properties

Label 4-3920e2-1.1-c1e2-0-3
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $979.774$
Root an. cond. $5.59476$
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  − 2·3-s − 2·5-s − 9-s + 4·11-s − 4·13-s + 4·15-s + 4·17-s + 14·23-s + 3·25-s + 6·27-s − 10·29-s − 4·31-s − 8·33-s + 8·39-s − 6·41-s − 6·43-s + 2·45-s + 12·47-s − 8·51-s − 8·55-s − 8·59-s − 2·61-s + 8·65-s + 6·67-s − 28·69-s + 24·71-s + 4·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 1/3·9-s + 1.20·11-s − 1.10·13-s + 1.03·15-s + 0.970·17-s + 2.91·23-s + 3/5·25-s + 1.15·27-s − 1.85·29-s − 0.718·31-s − 1.39·33-s + 1.28·39-s − 0.937·41-s − 0.914·43-s + 0.298·45-s + 1.75·47-s − 1.12·51-s − 1.07·55-s − 1.04·59-s − 0.256·61-s + 0.992·65-s + 0.733·67-s − 3.37·69-s + 2.84·71-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(979.774\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.387271407\)
\(L(\frac12)\) \(\approx\) \(1.387271407\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 14 T + 93 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T - 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 2 T + 91 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 18 T + 229 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 22 T + 3 p T^{2} - 22 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−8.606639025731844223129765003900, −8.404790975608112696899972413596, −7.73974977323421274206596986501, −7.38067759437325940149333477478, −7.24097110621780434150306656115, −6.89598350563365774795898987171, −6.38859648762189023262285128765, −6.14017434163924361201237611326, −5.51714118514800690938145080171, −5.28601376554767630912251357432, −4.98155971055383183762337452585, −4.72747925765748223803446300538, −3.98240485332028859171618924374, −3.70619228927635773217993184363, −3.15033913867567121032627451940, −3.02586397501364634944670021640, −2.13060140297231017883952378822, −1.64124097874298841744850039211, −0.67142333828746085447206645163, −0.62526435522721393061037845624, 0.62526435522721393061037845624, 0.67142333828746085447206645163, 1.64124097874298841744850039211, 2.13060140297231017883952378822, 3.02586397501364634944670021640, 3.15033913867567121032627451940, 3.70619228927635773217993184363, 3.98240485332028859171618924374, 4.72747925765748223803446300538, 4.98155971055383183762337452585, 5.28601376554767630912251357432, 5.51714118514800690938145080171, 6.14017434163924361201237611326, 6.38859648762189023262285128765, 6.89598350563365774795898987171, 7.24097110621780434150306656115, 7.38067759437325940149333477478, 7.73974977323421274206596986501, 8.404790975608112696899972413596, 8.606639025731844223129765003900

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