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Properties

Label 4-85e2-1.1-c1e2-0-1
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s − 4-s − 2·5-s + 4·6-s + 6·7-s − 8·8-s + 2·9-s − 4·10-s − 6·11-s − 2·12-s + 12·14-s − 4·15-s − 7·16-s + 2·17-s + 4·18-s + 2·20-s + 12·21-s − 12·22-s − 2·23-s − 16·24-s − 25-s + 6·27-s − 6·28-s − 6·29-s − 8·30-s − 2·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s + 1.63·6-s + 2.26·7-s − 2.82·8-s + 2/3·9-s − 1.26·10-s − 1.80·11-s − 0.577·12-s + 3.20·14-s − 1.03·15-s − 7/4·16-s + 0.485·17-s + 0.942·18-s + 0.447·20-s + 2.61·21-s − 2.55·22-s − 0.417·23-s − 3.26·24-s − 1/5·25-s + 1.15·27-s − 1.13·28-s − 1.11·29-s − 1.46·30-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: \(0\)
Selberg data: \((4,\ 7225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.713521602\)
\(L(\frac12)\) \(\approx\) \(1.713521602\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
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

−14.40160283512726406922595361117, −14.01362185971356011139195805864, −13.70071605705661377357561342452, −13.10032362330901460964222778441, −12.38290871747372188836599527133, −12.31514992730239331153242712356, −11.41117665555159837062054382400, −10.93744248030338334125964702834, −10.19238182224511771261920467036, −9.355977995757028984368500482249, −8.568972956638657282222548941950, −8.462854333626999577868475916988, −7.65144474197002644281810748719, −7.58763626479385463945349547505, −5.79663938906687122063619373336, −5.27091516089608358425373647377, −4.72497226585874361076547094617, −4.16080075191035978763604078265, −3.43197126874855401031714599883, −2.44264460611605416164578916134, 2.44264460611605416164578916134, 3.43197126874855401031714599883, 4.16080075191035978763604078265, 4.72497226585874361076547094617, 5.27091516089608358425373647377, 5.79663938906687122063619373336, 7.58763626479385463945349547505, 7.65144474197002644281810748719, 8.462854333626999577868475916988, 8.568972956638657282222548941950, 9.355977995757028984368500482249, 10.19238182224511771261920467036, 10.93744248030338334125964702834, 11.41117665555159837062054382400, 12.31514992730239331153242712356, 12.38290871747372188836599527133, 13.10032362330901460964222778441, 13.70071605705661377357561342452, 14.01362185971356011139195805864, 14.40160283512726406922595361117

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