Properties

Label 4-75e2-1.1-c2e2-0-0
Degree $4$
Conductor $5625$
Sign $1$
Analytic cond. $4.17630$
Root an. cond. $1.42954$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s − 3·4-s + 16·9-s + 15·12-s − 20·13-s − 7·16-s + 14·19-s − 35·27-s + 84·31-s − 48·36-s + 80·37-s + 100·39-s + 100·43-s + 35·48-s − 98·49-s + 60·52-s − 70·57-s − 16·61-s + 69·64-s − 90·67-s + 70·73-s − 42·76-s + 24·79-s + 31·81-s − 420·93-s + 140·97-s + 140·103-s + ⋯
L(s)  = 1  − 5/3·3-s − 3/4·4-s + 16/9·9-s + 5/4·12-s − 1.53·13-s − 0.437·16-s + 0.736·19-s − 1.29·27-s + 2.70·31-s − 4/3·36-s + 2.16·37-s + 2.56·39-s + 2.32·43-s + 0.729·48-s − 2·49-s + 1.15·52-s − 1.22·57-s − 0.262·61-s + 1.07·64-s − 1.34·67-s + 0.958·73-s − 0.552·76-s + 0.303·79-s + 0.382·81-s − 4.51·93-s + 1.44·97-s + 1.35·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(4.17630\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{75} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5625,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5426514474\)
\(L(\frac12)\) \(\approx\) \(0.5426514474\)
\(L(2)\) 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)$
bad3$C_2$ \( 1 + 5 T + p^{2} T^{2} \)
5 \( 1 \)
good2$C_2^2$ \( 1 + 3 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 + 3 p T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 567 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 662 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 582 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3087 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2262 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 3462 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 2562 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 45 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 8982 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 35 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 8927 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 97 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 - 70 T + p^{2} T^{2} )^{2} \)
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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

−15.17504411102317779283812417356, −14.41943868067291434002100009519, −13.25897015978267317957162402064, −12.91579344036196336152965421658, −12.26707260963880281144351454594, −11.78371664995541256777090216274, −11.47019098443622841110943855599, −10.71170401709734369883665800005, −10.13071473858409237880329163208, −9.573970437635075759008966940938, −9.235359735038001133082341022135, −8.031786887265594906957224790805, −7.59111921228887151435526646833, −6.69220502956559773239392690377, −6.17892388696250017207784049152, −5.37585889208735346786348859705, −4.59375851392575445562101445835, −4.44837874626504781478251437398, −2.70426003703818706466712273760, −0.72507732653626953368632610725, 0.72507732653626953368632610725, 2.70426003703818706466712273760, 4.44837874626504781478251437398, 4.59375851392575445562101445835, 5.37585889208735346786348859705, 6.17892388696250017207784049152, 6.69220502956559773239392690377, 7.59111921228887151435526646833, 8.031786887265594906957224790805, 9.235359735038001133082341022135, 9.573970437635075759008966940938, 10.13071473858409237880329163208, 10.71170401709734369883665800005, 11.47019098443622841110943855599, 11.78371664995541256777090216274, 12.26707260963880281144351454594, 12.91579344036196336152965421658, 13.25897015978267317957162402064, 14.41943868067291434002100009519, 15.17504411102317779283812417356

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