Properties

Label 4-35e2-1.1-c5e2-0-0
Degree $4$
Conductor $1225$
Sign $1$
Analytic cond. $31.5106$
Root an. cond. $2.36926$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s − 47·4-s − 50·5-s + 3·6-s − 98·7-s − 63·8-s − 333·9-s − 50·10-s − 601·11-s − 141·12-s − 577·13-s − 98·14-s − 150·15-s + 1.20e3·16-s + 41·17-s − 333·18-s + 630·19-s + 2.35e3·20-s − 294·21-s − 601·22-s − 442·23-s − 189·24-s + 1.87e3·25-s − 577·26-s − 1.29e3·27-s + 4.60e3·28-s + ⋯
L(s)  = 1  + 0.176·2-s + 0.192·3-s − 1.46·4-s − 0.894·5-s + 0.0340·6-s − 0.755·7-s − 0.348·8-s − 1.37·9-s − 0.158·10-s − 1.49·11-s − 0.282·12-s − 0.946·13-s − 0.133·14-s − 0.172·15-s + 1.17·16-s + 0.0344·17-s − 0.242·18-s + 0.400·19-s + 1.31·20-s − 0.145·21-s − 0.264·22-s − 0.174·23-s − 0.0669·24-s + 3/5·25-s − 0.167·26-s − 0.342·27-s + 1.11·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.5106\)
Root analytic conductor: \(2.36926\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1225,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{2} \)
7$C_1$ \( ( 1 + p^{2} T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 p^{4} T^{2} - p^{5} T^{3} + p^{10} T^{4} \)
3$D_{4}$ \( 1 - p T + 38 p^{2} T^{2} - p^{6} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 601 T + 259506 T^{2} + 601 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 577 T + 780172 T^{2} + 577 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 41 T + 1816368 T^{2} - 41 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 630 T + 4931238 T^{2} - 630 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 442 T - 299538 T^{2} + 442 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 5885 T + 35168948 T^{2} - 5885 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 396 T + 43423646 T^{2} + 396 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 8904 T + 132709718 T^{2} + 8904 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 1774 T - 4379094 T^{2} - 1774 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 27122 T + 464103742 T^{2} + 27122 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 21289 T + 531685238 T^{2} + 21289 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 55582 T + 1605381282 T^{2} + 55582 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 59600 T + 1881188438 T^{2} - 59600 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 51846 T + 1832119946 T^{2} + 51846 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 45344 T + 2875187158 T^{2} + 45344 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 80744 T + 4781205326 T^{2} - 80744 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 13532 T + 3906362902 T^{2} + 13532 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 51795 T + 1771153398 T^{2} + 51795 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 109828 T + 10643414822 T^{2} - 109828 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 37650 T + 10990453658 T^{2} + 37650 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 96339 T + 16335863448 T^{2} + 96339 p^{5} T^{3} + p^{10} T^{4} \)
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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

−14.94808582298613532937978421080, −14.73441856505347053689854138156, −13.74170258001829973367498041817, −13.69119822940869710538499628832, −12.78101222722675327510565152481, −12.35807842413409799800317033984, −11.67654231930691529753016475398, −10.83543312987410794957736278271, −9.975824195018469546036007313656, −9.557668410900609873242043847864, −8.457254639796722827613336821948, −8.365465018255132161276631766095, −7.49083378784061391460195292367, −6.34587109037505171982306508693, −5.10370852580996760227812753283, −4.90416902353730511480470986203, −3.48907564507549800331574950178, −2.86905804352889549176612019967, 0, 0, 2.86905804352889549176612019967, 3.48907564507549800331574950178, 4.90416902353730511480470986203, 5.10370852580996760227812753283, 6.34587109037505171982306508693, 7.49083378784061391460195292367, 8.365465018255132161276631766095, 8.457254639796722827613336821948, 9.557668410900609873242043847864, 9.975824195018469546036007313656, 10.83543312987410794957736278271, 11.67654231930691529753016475398, 12.35807842413409799800317033984, 12.78101222722675327510565152481, 13.69119822940869710538499628832, 13.74170258001829973367498041817, 14.73441856505347053689854138156, 14.94808582298613532937978421080

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