Properties

Label 4-35e2-1.1-c2e2-0-1
Degree $4$
Conductor $1225$
Sign $1$
Analytic cond. $0.909507$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·4-s − 4·7-s − 16·8-s + 13·9-s − 2·11-s − 16·14-s − 64·16-s + 52·18-s − 8·22-s + 16·23-s − 5·25-s − 16·28-s + 82·29-s − 64·32-s + 52·36-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 − 52·63-s + 192·64-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 4/7·7-s − 2·8-s + 13/9·9-s − 0.181·11-s − 8/7·14-s − 4·16-s + 26/9·18-s − 0.363·22-s + 0.695·23-s − 1/5·25-s − 4/7·28-s + 2.82·29-s − 2·32-s + 13/9·36-s − 1.51·37-s − 3.81·43-s − 0.181·44-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 − 0.825·63-s + 3·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.909507\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1225,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.025287142\)
\(L(\frac12)\) \(\approx\) \(2.025287142\)
\(L(2)\) 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)$
bad5$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 + 4 T + p^{2} T^{2} \)
good2$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
3$C_2^2$ \( 1 - 13 T^{2} + p^{4} T^{4} \)
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

−16.19003335948102889435073738464, −15.78843731153310701546008509645, −15.10298044899458491757036242703, −14.95421579659256666946125634018, −13.86759591438522435402586277751, −13.70663786638314494099409636780, −13.08720305968395502503817355295, −12.71106252931408021522466085719, −12.03576960908349266341318005236, −11.70603651917400914192694043400, −10.11995477735037076388918724790, −10.11223005987859719346863047131, −8.957678794272703689224554903528, −8.355342312889916135652621542860, −6.75293607249556376196345518939, −6.61392551631443716343866201571, −5.27925486380411942242917969488, −4.79748046709348623827271120853, −3.91446018637379515645859447007, −3.05371396911243213951153324425, 3.05371396911243213951153324425, 3.91446018637379515645859447007, 4.79748046709348623827271120853, 5.27925486380411942242917969488, 6.61392551631443716343866201571, 6.75293607249556376196345518939, 8.355342312889916135652621542860, 8.957678794272703689224554903528, 10.11223005987859719346863047131, 10.11995477735037076388918724790, 11.70603651917400914192694043400, 12.03576960908349266341318005236, 12.71106252931408021522466085719, 13.08720305968395502503817355295, 13.70663786638314494099409636780, 13.86759591438522435402586277751, 14.95421579659256666946125634018, 15.10298044899458491757036242703, 15.78843731153310701546008509645, 16.19003335948102889435073738464

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