Properties

Label 4-300e2-1.1-c2e2-0-9
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 + 16·7-s + 7·9-s + 24·13-s + 12·19-s − 64·21-s + 8·27-s + 68·31-s + 88·37-s − 96·39-s − 56·43-s + 94·49-s − 48·57-s + 148·61-s + 112·63-s − 184·67-s + 112·73-s + 156·79-s − 95·81-s + 384·91-s − 272·93-s − 64·97-s − 208·103-s − 148·109-s − 352·111-s + 168·117-s + 162·121-s + ⋯
L(s)  = 1  − 4/3·3-s + 16/7·7-s + 7/9·9-s + 1.84·13-s + 0.631·19-s − 3.04·21-s + 8/27·27-s + 2.19·31-s + 2.37·37-s − 2.46·39-s − 1.30·43-s + 1.91·49-s − 0.842·57-s + 2.42·61-s + 16/9·63-s − 2.74·67-s + 1.53·73-s + 1.97·79-s − 1.17·81-s + 4.21·91-s − 2.92·93-s − 0.659·97-s − 2.01·103-s − 1.35·109-s − 3.17·111-s + 1.43·117-s + 1.33·121-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

\(L(\frac{3}{2})\) \(\approx\) \(2.251222914\)
\(L(\frac12)\) \(\approx\) \(2.251222914\)
\(L(2)\) 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$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 162 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 12 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 402 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 1038 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 962 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 44 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3042 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 28 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4398 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 3998 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 + 2718 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 74 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 92 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7202 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 56 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 78 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 3198 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 15522 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 32 T + p^{2} T^{2} )^{2} \)
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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

−11.63127316357116860112761318111, −11.26994784482319703813811730186, −10.96985493007450259523492845398, −10.72481616995928590502940866135, −9.980878689875229571322506235101, −9.570171244330608412569500349874, −8.693072291846072503169515899157, −8.259450971420396431104118184756, −8.095618040348254810084468408010, −7.51682248996287130635354395387, −6.54725523496534546920572089459, −6.42959859562964145080900312788, −5.63854645933505780650899994485, −5.32712756576990242069786800323, −4.52428353551788446656738845372, −4.49265375350118589065458508438, −3.47457750421027810608906245331, −2.41276010146875839090153387652, −1.28691995407705465160167878900, −1.00875094651662017529541759925, 1.00875094651662017529541759925, 1.28691995407705465160167878900, 2.41276010146875839090153387652, 3.47457750421027810608906245331, 4.49265375350118589065458508438, 4.52428353551788446656738845372, 5.32712756576990242069786800323, 5.63854645933505780650899994485, 6.42959859562964145080900312788, 6.54725523496534546920572089459, 7.51682248996287130635354395387, 8.095618040348254810084468408010, 8.259450971420396431104118184756, 8.693072291846072503169515899157, 9.570171244330608412569500349874, 9.980878689875229571322506235101, 10.72481616995928590502940866135, 10.96985493007450259523492845398, 11.26994784482319703813811730186, 11.63127316357116860112761318111

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