Properties

Label 4-35e2-1.1-c4e2-0-1
Degree $4$
Conductor $1225$
Sign $1$
Analytic cond. $13.0895$
Root an. cond. $1.90209$
Motivic weight $4$
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  + 10·3-s + 26·4-s + 10·5-s + 70·7-s − 87·9-s + 178·11-s + 260·12-s − 10·13-s + 100·15-s + 420·16-s − 970·17-s + 260·20-s + 700·21-s − 525·25-s − 1.93e3·27-s + 1.82e3·28-s + 382·29-s + 1.78e3·33-s + 700·35-s − 2.26e3·36-s − 100·39-s + 4.62e3·44-s − 870·45-s − 4.39e3·47-s + 4.20e3·48-s + 2.49e3·49-s − 9.70e3·51-s + ⋯
L(s)  = 1  + 10/9·3-s + 13/8·4-s + 2/5·5-s + 10/7·7-s − 1.07·9-s + 1.47·11-s + 1.80·12-s − 0.0591·13-s + 4/9·15-s + 1.64·16-s − 3.35·17-s + 0.649·20-s + 1.58·21-s − 0.839·25-s − 2.64·27-s + 2.32·28-s + 0.454·29-s + 1.63·33-s + 4/7·35-s − 1.74·36-s − 0.0657·39-s + 2.39·44-s − 0.429·45-s − 1.98·47-s + 1.82·48-s + 1.04·49-s − 3.72·51-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(13.0895\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1225,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.883736733\)
\(L(\frac12)\) \(\approx\) \(3.883736733\)
\(L(3)\) 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$C_2$ \( 1 - 2 p T + p^{4} T^{2} \)
7$C_2$ \( 1 - 10 p T + p^{4} T^{2} \)
good2$C_2^2$ \( 1 - 13 p T^{2} + p^{8} T^{4} \)
3$C_2$ \( ( 1 - 5 T + p^{4} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 89 T + p^{4} T^{2} )^{2} \)
13$C_2$ \( ( 1 + 5 T + p^{4} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 485 T + p^{4} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 212042 T^{2} + p^{8} T^{4} \)
23$C_2^2$ \( 1 - 68906 T^{2} + p^{8} T^{4} \)
29$C_2$ \( ( 1 - 191 T + p^{4} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 737642 T^{2} + p^{8} T^{4} \)
37$C_2^2$ \( 1 - 794 p^{2} T^{2} + p^{8} T^{4} \)
41$C_2^2$ \( 1 + 2845078 T^{2} + p^{8} T^{4} \)
43$C_2^2$ \( 1 - 6695306 T^{2} + p^{8} T^{4} \)
47$C_2$ \( ( 1 + 2195 T + p^{4} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 13261538 T^{2} + p^{8} T^{4} \)
59$C_2^2$ \( 1 - 11092322 T^{2} + p^{8} T^{4} \)
61$C_2^2$ \( 1 - 23947082 T^{2} + p^{8} T^{4} \)
67$C_2^2$ \( 1 - 36108866 T^{2} + p^{8} T^{4} \)
71$C_2$ \( ( 1 - 4454 T + p^{4} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 8650 T + p^{4} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 5561 T + p^{4} T^{2} )^{2} \)
83$C_2$ \( ( 1 - 1990 T + p^{4} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 124831082 T^{2} + p^{8} T^{4} \)
97$C_2$ \( ( 1 - 9235 T + p^{4} T^{2} )^{2} \)
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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

−15.78605964050010973043571338674, −15.36150055342789548375809918933, −14.85273661187535033468194370028, −14.45065008951463534921524175665, −13.82124915359663082856912551027, −13.42837995572760950408359374228, −12.20360554861326034382482696112, −11.51230780852540892505119352806, −11.15109951002768807288444074118, −11.00590441988255020458470577646, −9.558580304904769233577668359034, −8.905162000563469516246488216410, −8.372013421659663015342318746996, −7.73872135544898343163271737015, −6.49968452293574160235912884544, −6.41831416280636474931610874263, −4.96861067944583778768211865705, −3.67335978508331130762491268181, −2.21157011314326723139645887774, −2.07092370770431135743982480020, 2.07092370770431135743982480020, 2.21157011314326723139645887774, 3.67335978508331130762491268181, 4.96861067944583778768211865705, 6.41831416280636474931610874263, 6.49968452293574160235912884544, 7.73872135544898343163271737015, 8.372013421659663015342318746996, 8.905162000563469516246488216410, 9.558580304904769233577668359034, 11.00590441988255020458470577646, 11.15109951002768807288444074118, 11.51230780852540892505119352806, 12.20360554861326034382482696112, 13.42837995572760950408359374228, 13.82124915359663082856912551027, 14.45065008951463534921524175665, 14.85273661187535033468194370028, 15.36150055342789548375809918933, 15.78605964050010973043571338674

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