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Properties

Label 4-85e2-1.1-c1e2-0-3
Degree $4$
Conductor $7225$
Sign $1$
Analytic cond. $0.460672$
Root an. cond. $0.823849$
Motivic weight $1$
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  − 2·2-s − 4·3-s + 4-s − 2·5-s + 8·6-s − 4·7-s + 8·9-s + 4·10-s − 8·11-s − 4·12-s + 8·14-s + 8·15-s + 16-s − 2·17-s − 16·18-s − 2·20-s + 16·21-s + 16·22-s − 4·23-s + 3·25-s − 12·27-s − 4·28-s − 4·29-s − 16·30-s + 2·32-s + 32·33-s + 4·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s + 3.26·6-s − 1.51·7-s + 8/3·9-s + 1.26·10-s − 2.41·11-s − 1.15·12-s + 2.13·14-s + 2.06·15-s + 1/4·16-s − 0.485·17-s − 3.77·18-s − 0.447·20-s + 3.49·21-s + 3.41·22-s − 0.834·23-s + 3/5·25-s − 2.30·27-s − 0.755·28-s − 0.742·29-s − 2.92·30-s + 0.353·32-s + 5.57·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7225\)    =    \(5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.460672\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 24 T + 254 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 172 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−13.58208057480413899838428201851, −13.26182409131366112701738937553, −12.57805276161533657208570884782, −12.31775539685803957797677093771, −11.72015210452327744400488944927, −11.06308369725245175872328519507, −10.57847569920422134851528053543, −10.39126818503970461796256783247, −9.653658474315122243342761418134, −9.203714202944556418782199011166, −8.223352963482811590314374917504, −7.75778713540626957529223558073, −7.10272384563967549403206342880, −6.31406013723778519305150017524, −5.77708222231259154740652373264, −5.19016061384923701683556479987, −4.27247643895843496960279170082, −2.95634545112088224055144157853, 0, 0, 2.95634545112088224055144157853, 4.27247643895843496960279170082, 5.19016061384923701683556479987, 5.77708222231259154740652373264, 6.31406013723778519305150017524, 7.10272384563967549403206342880, 7.75778713540626957529223558073, 8.223352963482811590314374917504, 9.203714202944556418782199011166, 9.653658474315122243342761418134, 10.39126818503970461796256783247, 10.57847569920422134851528053543, 11.06308369725245175872328519507, 11.72015210452327744400488944927, 12.31775539685803957797677093771, 12.57805276161533657208570884782, 13.26182409131366112701738937553, 13.58208057480413899838428201851

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