Properties

Label 4-300e2-1.1-c1e2-0-5
Degree $4$
Conductor $90000$
Sign $1$
Analytic cond. $5.73847$
Root an. cond. $1.54774$
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  + 2·2-s + 2·4-s + 9-s + 2·13-s − 4·16-s + 4·17-s + 2·18-s + 4·26-s + 20·29-s − 8·32-s + 8·34-s + 2·36-s + 4·37-s − 16·41-s − 5·49-s + 4·52-s − 8·53-s + 40·58-s + 14·61-s − 8·64-s + 8·68-s − 28·73-s + 8·74-s + 81-s − 32·82-s + 34·97-s − 10·98-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1/3·9-s + 0.554·13-s − 16-s + 0.970·17-s + 0.471·18-s + 0.784·26-s + 3.71·29-s − 1.41·32-s + 1.37·34-s + 1/3·36-s + 0.657·37-s − 2.49·41-s − 5/7·49-s + 0.554·52-s − 1.09·53-s + 5.25·58-s + 1.79·61-s − 64-s + 0.970·68-s − 3.27·73-s + 0.929·74-s + 1/9·81-s − 3.53·82-s + 3.45·97-s − 1.01·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(90000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5.73847\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{90000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 90000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.274603091\)
\(L(\frac12)\) \(\approx\) \(3.274603091\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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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

−9.946237306464798320060676890295, −8.979440455637625961269202062502, −8.686228257120952969207606021954, −8.134034752267247325522916900531, −7.57404393602003080106109524301, −6.79375443883521821659151920822, −6.47388999507393793454845833318, −6.05569215012514529764032801313, −5.34490029281998815763412767656, −4.63718444572594903228967418102, −4.62309319523854591570832526017, −3.50818980571584578672789924466, −3.24466149791278329969031792942, −2.44041792194616260799140408514, −1.24790523142424415923064013417, 1.24790523142424415923064013417, 2.44041792194616260799140408514, 3.24466149791278329969031792942, 3.50818980571584578672789924466, 4.62309319523854591570832526017, 4.63718444572594903228967418102, 5.34490029281998815763412767656, 6.05569215012514529764032801313, 6.47388999507393793454845833318, 6.79375443883521821659151920822, 7.57404393602003080106109524301, 8.134034752267247325522916900531, 8.686228257120952969207606021954, 8.979440455637625961269202062502, 9.946237306464798320060676890295

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