Properties

Label 4-1850e2-1.1-c1e2-0-20
Degree $4$
Conductor $3422500$
Sign $1$
Analytic cond. $218.221$
Root an. cond. $3.84347$
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  + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 6·7-s + 4·8-s − 4·11-s + 6·12-s + 4·13-s + 12·14-s + 5·16-s − 4·17-s + 2·19-s + 12·21-s − 8·22-s + 16·23-s + 8·24-s + 8·26-s − 2·27-s + 18·28-s − 4·29-s − 2·31-s + 6·32-s − 8·33-s − 8·34-s − 2·37-s + 4·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 2.26·7-s + 1.41·8-s − 1.20·11-s + 1.73·12-s + 1.10·13-s + 3.20·14-s + 5/4·16-s − 0.970·17-s + 0.458·19-s + 2.61·21-s − 1.70·22-s + 3.33·23-s + 1.63·24-s + 1.56·26-s − 0.384·27-s + 3.40·28-s − 0.742·29-s − 0.359·31-s + 1.06·32-s − 1.39·33-s − 1.37·34-s − 0.328·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3422500\)    =    \(2^{2} \cdot 5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(218.221\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(13.66459957\)
\(L(\frac12)\) \(\approx\) \(13.66459957\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 14 T + 204 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 212 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−9.004001430377689944915927007538, −8.963076370905079983796561663384, −8.598733423047630184155344837903, −8.357825002388308440132058755782, −7.72120075724208239820633257686, −7.47499756616071603867196585268, −7.23218117195997713434059500337, −6.75436080125806691900573851388, −5.99270558720571648066210713444, −5.73891220942164057182818500067, −5.08025389328291495328963856726, −4.95707356291623409317770773447, −4.70847845368273107965005237560, −4.09544556474249645892883802569, −3.34847261523396353140969350432, −3.31006486743633003984378411175, −2.45575591700564970785754356198, −2.39039497173098737359083859503, −1.57118002568879529537407800860, −1.10706962114590556341978402535, 1.10706962114590556341978402535, 1.57118002568879529537407800860, 2.39039497173098737359083859503, 2.45575591700564970785754356198, 3.31006486743633003984378411175, 3.34847261523396353140969350432, 4.09544556474249645892883802569, 4.70847845368273107965005237560, 4.95707356291623409317770773447, 5.08025389328291495328963856726, 5.73891220942164057182818500067, 5.99270558720571648066210713444, 6.75436080125806691900573851388, 7.23218117195997713434059500337, 7.47499756616071603867196585268, 7.72120075724208239820633257686, 8.357825002388308440132058755782, 8.598733423047630184155344837903, 8.963076370905079983796561663384, 9.004001430377689944915927007538

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