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Properties

Label 4-85e2-1.1-c1e2-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s − 4-s − 4·5-s + 4·6-s − 6·7-s + 8·8-s + 2·9-s + 8·10-s − 6·11-s + 2·12-s + 12·14-s + 8·15-s − 7·16-s − 2·17-s − 4·18-s + 4·20-s + 12·21-s + 12·22-s + 2·23-s − 16·24-s + 11·25-s − 6·27-s + 6·28-s − 6·29-s − 16·30-s − 2·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s − 1/2·4-s − 1.78·5-s + 1.63·6-s − 2.26·7-s + 2.82·8-s + 2/3·9-s + 2.52·10-s − 1.80·11-s + 0.577·12-s + 3.20·14-s + 2.06·15-s − 7/4·16-s − 0.485·17-s − 0.942·18-s + 0.894·20-s + 2.61·21-s + 2.55·22-s + 0.417·23-s − 3.26·24-s + 11/5·25-s − 1.15·27-s + 1.13·28-s − 1.11·29-s − 2.92·30-s − 0.359·31-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$C_2$ \( 1 + 4 T + p T^{2} \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−13.42611989823108419734907230713, −13.32630533814246610775881387999, −12.94830783645454108530989986535, −12.48684728758773046959987897227, −11.59762270034917081739924900375, −11.18274439729825220667769445726, −10.32599067382036335971290078469, −10.32552968461798858128312043103, −9.536000449672005898168666201275, −9.119068847231446390664997108869, −8.097617687612107227145122256818, −8.085648823994533951087094693706, −7.09993871166123286783735789360, −6.79747013306469590511452414662, −5.50118783410996279208378411556, −4.95977831271023141004118810141, −4.01696982023488393293456825090, −3.32137479342394424359904830677, 0, 0, 3.32137479342394424359904830677, 4.01696982023488393293456825090, 4.95977831271023141004118810141, 5.50118783410996279208378411556, 6.79747013306469590511452414662, 7.09993871166123286783735789360, 8.085648823994533951087094693706, 8.097617687612107227145122256818, 9.119068847231446390664997108869, 9.536000449672005898168666201275, 10.32552968461798858128312043103, 10.32599067382036335971290078469, 11.18274439729825220667769445726, 11.59762270034917081739924900375, 12.48684728758773046959987897227, 12.94830783645454108530989986535, 13.32630533814246610775881387999, 13.42611989823108419734907230713

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