Properties

Label 4-300e2-1.1-c2e2-0-0
Degree $4$
Conductor $90000$
Sign $1$
Analytic cond. $66.8209$
Root an. cond. $2.85909$
Motivic weight $2$
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  − 4·3-s − 4·7-s + 7·9-s − 16·13-s − 68·19-s + 16·21-s + 8·27-s + 28·31-s − 112·37-s + 64·39-s − 16·43-s − 86·49-s + 272·57-s − 92·61-s − 28·63-s − 64·67-s + 212·73-s − 44·79-s − 95·81-s + 64·91-s − 112·93-s − 244·97-s + 92·103-s + 172·109-s + 448·111-s − 112·117-s + 62·121-s + ⋯
L(s)  = 1  − 4/3·3-s − 4/7·7-s + 7/9·9-s − 1.23·13-s − 3.57·19-s + 0.761·21-s + 8/27·27-s + 0.903·31-s − 3.02·37-s + 1.64·39-s − 0.372·43-s − 1.75·49-s + 4.77·57-s − 1.50·61-s − 4/9·63-s − 0.955·67-s + 2.90·73-s − 0.556·79-s − 1.17·81-s + 0.703·91-s − 1.20·93-s − 2.51·97-s + 0.893·103-s + 1.57·109-s + 4.03·111-s − 0.957·117-s + 0.512·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(90000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(66.8209\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 90000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01474831082\)
\(L(\frac12)\) \(\approx\) \(0.01474831082\)
\(L(2)\) 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$C_2$ \( 1 + 4 T + p^{2} T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 62 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 398 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 34 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 562 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 62 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 56 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2642 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2798 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 3998 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6782 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 32 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7202 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 106 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 802 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 + 122 T + p^{2} T^{2} )^{2} \)
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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

−11.87864734125705491837559771716, −11.03348689945468347922973209114, −10.99188207958482789277875478884, −10.27625407967220984240041654128, −10.21864449823161517679693539571, −9.565093579568376055727374681660, −8.845963533761795134348908928635, −8.476213058343883407330887129666, −7.999983778812660835548324606786, −7.03090524754265786503083261382, −6.83877511425849965420923513757, −6.24466566045780383254682816019, −6.05027237244005334117503226722, −5.03066604984611182841641545587, −4.85754041127780784443162331924, −4.21209483508556133464090483093, −3.40651597295852309092673409727, −2.46400498127715373100561176813, −1.73456102804551241800635838660, −0.06113377302895533249386842189, 0.06113377302895533249386842189, 1.73456102804551241800635838660, 2.46400498127715373100561176813, 3.40651597295852309092673409727, 4.21209483508556133464090483093, 4.85754041127780784443162331924, 5.03066604984611182841641545587, 6.05027237244005334117503226722, 6.24466566045780383254682816019, 6.83877511425849965420923513757, 7.03090524754265786503083261382, 7.999983778812660835548324606786, 8.476213058343883407330887129666, 8.845963533761795134348908928635, 9.565093579568376055727374681660, 10.21864449823161517679693539571, 10.27625407967220984240041654128, 10.99188207958482789277875478884, 11.03348689945468347922973209114, 11.87864734125705491837559771716

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