Properties

Label 4-650e2-1.1-c1e2-0-2
Degree $4$
Conductor $422500$
Sign $1$
Analytic cond. $26.9389$
Root an. cond. $2.27821$
Motivic weight $1$
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  − 4-s − 3·9-s − 4·11-s + 16-s − 12·19-s − 4·29-s + 8·31-s + 3·36-s + 4·44-s + 13·49-s + 20·59-s − 16·61-s − 64-s − 10·71-s + 12·76-s + 8·79-s − 12·89-s + 12·99-s + 8·101-s − 38·109-s + 4·116-s − 10·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 9-s − 1.20·11-s + 1/4·16-s − 2.75·19-s − 0.742·29-s + 1.43·31-s + 1/2·36-s + 0.603·44-s + 13/7·49-s + 2.60·59-s − 2.04·61-s − 1/8·64-s − 1.18·71-s + 1.37·76-s + 0.900·79-s − 1.27·89-s + 1.20·99-s + 0.796·101-s − 3.63·109-s + 0.371·116-s − 0.909·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(422500\)    =    \(2^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(26.9389\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6509244713\)
\(L(\frac12)\) \(\approx\) \(0.6509244713\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 75 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−10.92212570224779548370791656818, −10.34637589690869433857659804357, −10.06799930637953362653632943594, −9.413787163724293978001974088461, −8.809820151567755107786933907343, −8.694942736959701677206179958993, −8.238944565225757111433706897891, −7.84178587428531036418788541081, −7.37112580201007436919039948363, −6.48991124556445382873968124897, −6.47535775770838885116557535044, −5.57145826410981963624234480111, −5.50979829630104000635195324897, −4.78194685621167123840637861568, −4.13922793122795830586748637003, −3.95310562886591012285785759728, −2.74242594259921622419781338528, −2.70390667021701711718861425741, −1.82757827488921916657615179461, −0.42049051145004238766916968692, 0.42049051145004238766916968692, 1.82757827488921916657615179461, 2.70390667021701711718861425741, 2.74242594259921622419781338528, 3.95310562886591012285785759728, 4.13922793122795830586748637003, 4.78194685621167123840637861568, 5.50979829630104000635195324897, 5.57145826410981963624234480111, 6.47535775770838885116557535044, 6.48991124556445382873968124897, 7.37112580201007436919039948363, 7.84178587428531036418788541081, 8.238944565225757111433706897891, 8.694942736959701677206179958993, 8.809820151567755107786933907343, 9.413787163724293978001974088461, 10.06799930637953362653632943594, 10.34637589690869433857659804357, 10.92212570224779548370791656818

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