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Properties

Label 4-68e2-1.1-c0e2-0-0
Degree $4$
Conductor $4624$
Sign $1$
Analytic cond. $0.00115168$
Root an. cond. $0.184218$
Motivic weight $0$
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  − 4-s − 2·5-s + 16-s − 2·17-s + 2·20-s + 2·25-s + 2·29-s − 2·37-s + 2·41-s − 2·61-s − 64-s + 2·68-s + 2·73-s − 2·80-s − 81-s + 4·85-s − 2·97-s − 2·100-s − 2·109-s − 2·113-s − 2·116-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯
L(s)  = 1  − 4-s − 2·5-s + 16-s − 2·17-s + 2·20-s + 2·25-s + 2·29-s − 2·37-s + 2·41-s − 2·61-s − 64-s + 2·68-s + 2·73-s − 2·80-s − 81-s + 4·85-s − 2·97-s − 2·100-s − 2·109-s − 2·113-s − 2·116-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4624\)    =    \(2^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.00115168\)
Root analytic conductor: \(0.184218\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4624,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1773206590\)
\(L(\frac12)\) \(\approx\) \(0.1773206590\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−15.59461919004984204952321323569, −14.84625762244638749011823645362, −14.21237496700354657276706814216, −13.73708928432188607873060131214, −13.15884627493339383429814612152, −12.36784286101715037183258870678, −12.21051665090741868031115105838, −11.54553781733229543264173850657, −10.68372419403733204059333291826, −10.67475820275141773886786578035, −9.401820822599483704682833492791, −8.986147440681671677679816455417, −8.208046929617629467128032615792, −8.063910698916515610767798864707, −7.09193503008472694559106097842, −6.52495137986341477590907454457, −5.27019738107696468255483049268, −4.29878395807902601847606510864, −4.22564061217201658866752193569, −3.03544157255568814852703229915, 3.03544157255568814852703229915, 4.22564061217201658866752193569, 4.29878395807902601847606510864, 5.27019738107696468255483049268, 6.52495137986341477590907454457, 7.09193503008472694559106097842, 8.063910698916515610767798864707, 8.208046929617629467128032615792, 8.986147440681671677679816455417, 9.401820822599483704682833492791, 10.67475820275141773886786578035, 10.68372419403733204059333291826, 11.54553781733229543264173850657, 12.21051665090741868031115105838, 12.36784286101715037183258870678, 13.15884627493339383429814612152, 13.73708928432188607873060131214, 14.21237496700354657276706814216, 14.84625762244638749011823645362, 15.59461919004984204952321323569

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