Properties

Label 4-300e2-1.1-c2e2-0-7
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 + 36·13-s − 16·16-s + 20·17-s − 6·18-s + 72·26-s − 72·29-s + 40·34-s + 108·37-s + 36·41-s − 10·49-s − 52·53-s − 144·58-s − 148·61-s + 64·64-s + 24·72-s + 72·73-s + 216·74-s + 9·81-s + 72·82-s − 36·89-s − 144·97-s − 20·98-s + 72·101-s − 288·104-s + ⋯
L(s)  = 1  + 2-s − 8-s − 1/3·9-s + 2.76·13-s − 16-s + 1.17·17-s − 1/3·18-s + 2.76·26-s − 2.48·29-s + 1.17·34-s + 2.91·37-s + 0.878·41-s − 0.204·49-s − 0.981·53-s − 2.48·58-s − 2.42·61-s + 64-s + 1/3·72-s + 0.986·73-s + 2.91·74-s + 1/9·81-s + 0.878·82-s − 0.404·89-s − 1.48·97-s − 0.204·98-s + 0.712·101-s − 2.76·104-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

\(L(\frac{3}{2})\) \(\approx\) \(3.346887838\)
\(L(\frac12)\) \(\approx\) \(3.346887838\)
\(L(2)\) 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$C_2^2$ \( 1 + 10 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 134 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 530 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 1010 T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 + 36 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1874 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 - 54 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 3266 T^{2} + p^{4} T^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 5990 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 7250 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 + 718 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 36 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 4370 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 5666 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 72 T + p^{2} T^{2} )^{2} \)
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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

−11.60234847954900137159105801100, −11.44641659161806970646949378119, −10.84868311889043295451243031892, −10.76020046130107952399319556028, −9.687317209334965565593538840097, −9.256410222527187871226993973719, −9.185844305479484554992814331839, −8.222119278958223583640863311540, −8.074054975502293242663106833871, −7.47795926774721656314499227414, −6.54122075787208288844131846417, −6.11657885591481747662234166419, −5.68917968192121087710075349027, −5.50416165200524974151139829966, −4.26911001390627901531722190622, −4.21367817798966557315072725753, −3.27355068860280451216097271988, −3.13068473769664397930115799277, −1.79625612887449332423523611722, −0.812249866827834641146803614656, 0.812249866827834641146803614656, 1.79625612887449332423523611722, 3.13068473769664397930115799277, 3.27355068860280451216097271988, 4.21367817798966557315072725753, 4.26911001390627901531722190622, 5.50416165200524974151139829966, 5.68917968192121087710075349027, 6.11657885591481747662234166419, 6.54122075787208288844131846417, 7.47795926774721656314499227414, 8.074054975502293242663106833871, 8.222119278958223583640863311540, 9.185844305479484554992814331839, 9.256410222527187871226993973719, 9.687317209334965565593538840097, 10.76020046130107952399319556028, 10.84868311889043295451243031892, 11.44641659161806970646949378119, 11.60234847954900137159105801100

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