Properties

Label 4-300e2-1.1-c1e2-0-2
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-s − 4-s − 3·8-s + 9-s + 4·13-s − 16-s − 4·17-s + 18-s + 4·26-s − 4·29-s + 5·32-s − 4·34-s − 36-s + 20·37-s + 20·41-s − 14·49-s − 4·52-s + 20·53-s − 4·58-s − 4·61-s + 7·64-s + 4·68-s − 3·72-s − 20·73-s + 20·74-s + 81-s + 20·82-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1/3·9-s + 1.10·13-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.784·26-s − 0.742·29-s + 0.883·32-s − 0.685·34-s − 1/6·36-s + 3.28·37-s + 3.12·41-s − 2·49-s − 0.554·52-s + 2.74·53-s − 0.525·58-s − 0.512·61-s + 7/8·64-s + 0.485·68-s − 0.353·72-s − 2.34·73-s + 2.32·74-s + 1/9·81-s + 2.20·82-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

\(L(1)\) \(\approx\) \(1.788561370\)
\(L(\frac12)\) \(\approx\) \(1.788561370\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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

−9.655893575783761174602693945437, −9.049118048765622993735733025092, −8.843956595114680827452383410006, −8.151425844540532890281788534319, −7.68003844385765985083943135134, −7.10478788988449333222322211892, −6.34199635143511529870167101224, −5.86497653842965752491328535183, −5.72078508715635747921690343280, −4.52160444884874669934496061646, −4.42801570520906772580093162574, −3.86223113294681320091712334301, −3.01549784140686353642765162463, −2.32158421366848578869138486097, −0.957897388351054006923114027189, 0.957897388351054006923114027189, 2.32158421366848578869138486097, 3.01549784140686353642765162463, 3.86223113294681320091712334301, 4.42801570520906772580093162574, 4.52160444884874669934496061646, 5.72078508715635747921690343280, 5.86497653842965752491328535183, 6.34199635143511529870167101224, 7.10478788988449333222322211892, 7.68003844385765985083943135134, 8.151425844540532890281788534319, 8.843956595114680827452383410006, 9.049118048765622993735733025092, 9.655893575783761174602693945437

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