Properties

Label 4-300e2-1.1-c2e2-0-5
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  + 2·2-s − 8·8-s − 3·9-s − 4·13-s − 16·16-s − 20·17-s − 6·18-s − 8·26-s − 52·29-s − 40·34-s − 52·37-s + 116·41-s + 50·49-s + 148·53-s − 104·58-s + 52·61-s + 64·64-s + 24·72-s + 92·73-s − 104·74-s + 9·81-s + 232·82-s + 164·89-s − 4·97-s + 100·98-s − 148·101-s + 32·104-s + ⋯
L(s)  = 1  + 2-s − 8-s − 1/3·9-s − 0.307·13-s − 16-s − 1.17·17-s − 1/3·18-s − 0.307·26-s − 1.79·29-s − 1.17·34-s − 1.40·37-s + 2.82·41-s + 1.02·49-s + 2.79·53-s − 1.79·58-s + 0.852·61-s + 64-s + 1/3·72-s + 1.26·73-s − 1.40·74-s + 1/9·81-s + 2.82·82-s + 1.84·89-s − 0.0412·97-s + 1.02·98-s − 1.46·101-s + 4/13·104-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

\(L(\frac{3}{2})\) \(\approx\) \(2.037301914\)
\(L(\frac12)\) \(\approx\) \(2.037301914\)
\(L(2)\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 50 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 194 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1874 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 58 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 1346 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 382 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - 74 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 1150 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 8930 T^{2} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 - 46 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 1390 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 11426 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
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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

−12.20676232316273261844479902413, −11.37318921934460080913605010240, −10.86914994374009805816159690988, −10.62299954526491796619537476218, −9.721965955461200937777429923581, −9.435471181361320924532763970298, −8.830925477234147926690657007488, −8.649089061133141673114294565510, −7.88570578011239577205143688142, −7.09945477237034389091704740299, −7.02264388070960045120396929350, −5.95593084534368940392427728760, −5.85771213968725095321811117243, −5.17563019302033378856878539143, −4.64470667236947949773624715748, −3.86618194896178141329954625893, −3.71325821038048273749213094508, −2.56556922794828803380517569702, −2.19432048161390555066076840696, −0.57865852328494444581135371352, 0.57865852328494444581135371352, 2.19432048161390555066076840696, 2.56556922794828803380517569702, 3.71325821038048273749213094508, 3.86618194896178141329954625893, 4.64470667236947949773624715748, 5.17563019302033378856878539143, 5.85771213968725095321811117243, 5.95593084534368940392427728760, 7.02264388070960045120396929350, 7.09945477237034389091704740299, 7.88570578011239577205143688142, 8.649089061133141673114294565510, 8.830925477234147926690657007488, 9.435471181361320924532763970298, 9.721965955461200937777429923581, 10.62299954526491796619537476218, 10.86914994374009805816159690988, 11.37318921934460080913605010240, 12.20676232316273261844479902413

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