Properties

Label 4-315e2-1.1-c2e2-0-0
Degree $4$
Conductor $99225$
Sign $1$
Analytic cond. $73.6700$
Root an. cond. $2.92969$
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  − 4·2-s + 4·4-s − 4·7-s + 16·8-s + 2·11-s + 16·14-s − 64·16-s − 8·22-s − 16·23-s − 5·25-s − 16·28-s − 82·29-s + 64·32-s − 56·37-s − 164·43-s + 8·44-s + 64·46-s − 33·49-s + 20·50-s − 148·53-s − 64·56-s + 328·58-s + 192·64-s + 4·67-s − 28·71-s + 224·74-s − 8·77-s + ⋯
L(s)  = 1  − 2·2-s + 4-s − 4/7·7-s + 2·8-s + 2/11·11-s + 8/7·14-s − 4·16-s − 0.363·22-s − 0.695·23-s − 1/5·25-s − 4/7·28-s − 2.82·29-s + 2·32-s − 1.51·37-s − 3.81·43-s + 2/11·44-s + 1.39·46-s − 0.673·49-s + 2/5·50-s − 2.79·53-s − 8/7·56-s + 5.65·58-s + 3·64-s + 4/67·67-s − 0.394·71-s + 3.02·74-s − 0.103·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(99225\)    =    \(3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(73.6700\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 99225,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01132069763\)
\(L(\frac12)\) \(\approx\) \(0.01132069763\)
\(L(2)\) 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)$
bad3 \( 1 \)
5$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 + 4 T + p^{2} T^{2} \)
good2$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 67 T^{2} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 533 T^{2} + p^{4} T^{4} \)
19$C_2^2$ \( 1 - 542 T^{2} + p^{4} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 41 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 302 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 28 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3182 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4013 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 1858 T^{2} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 962 T^{2} + p^{4} T^{4} \)
67$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6158 T^{2} + p^{4} T^{4} \)
79$C_2$ \( ( 1 + 19 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4958 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \)
97$C_2^2$ \( 1 - 15173 T^{2} + p^{4} T^{4} \)
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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

−11.84203424991169838271414540858, −10.97741909588005109914613529969, −10.54553527187164359757110465447, −9.953496362739849879256930202707, −9.727827629813074115294914447336, −9.456194152073651115681409118674, −8.895942094570851454762338479792, −8.372050483188912683321698182252, −8.195361890838903567322890191856, −7.43392777070978793591519495422, −7.23775219911217240731815787989, −6.55934953317910651612077696776, −5.92056777180052733090108045347, −5.02279514388090544641428924112, −4.76048845611138046000366308947, −3.72714961700696734284648483796, −3.41737939146948647881093911051, −1.80452642677742445078820855679, −1.65480455906079568924477226587, −0.07209804574946439744557187539, 0.07209804574946439744557187539, 1.65480455906079568924477226587, 1.80452642677742445078820855679, 3.41737939146948647881093911051, 3.72714961700696734284648483796, 4.76048845611138046000366308947, 5.02279514388090544641428924112, 5.92056777180052733090108045347, 6.55934953317910651612077696776, 7.23775219911217240731815787989, 7.43392777070978793591519495422, 8.195361890838903567322890191856, 8.372050483188912683321698182252, 8.895942094570851454762338479792, 9.456194152073651115681409118674, 9.727827629813074115294914447336, 9.953496362739849879256930202707, 10.54553527187164359757110465447, 10.97741909588005109914613529969, 11.84203424991169838271414540858

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