Properties

Degree $4$
Conductor $19448100$
Sign $1$
Motivic weight $1$
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 + 3·4-s + 2·5-s − 4·8-s − 4·10-s + 4·11-s + 8·13-s + 5·16-s − 4·17-s + 4·19-s + 6·20-s − 8·22-s + 4·23-s + 3·25-s − 16·26-s + 8·29-s + 12·31-s − 6·32-s + 8·34-s − 4·37-s − 8·38-s − 8·40-s − 4·41-s + 12·44-s − 8·46-s − 8·47-s − 6·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.41·8-s − 1.26·10-s + 1.20·11-s + 2.21·13-s + 5/4·16-s − 0.970·17-s + 0.917·19-s + 1.34·20-s − 1.70·22-s + 0.834·23-s + 3/5·25-s − 3.13·26-s + 1.48·29-s + 2.15·31-s − 1.06·32-s + 1.37·34-s − 0.657·37-s − 1.29·38-s − 1.26·40-s − 0.624·41-s + 1.80·44-s − 1.17·46-s − 1.16·47-s − 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19448100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{4410} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19448100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.006043967\)
\(L(\frac12)\) \(\approx\) \(3.006043967\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good11$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−8.465464335452150337620407229382, −8.438068326580670186621370028747, −7.85159449087238944330398236164, −7.76982738325878983611731111009, −6.72180655486050609785172066075, −6.65940982850292086499004836608, −6.50217067955517898685784948115, −6.49290243910954680031437989975, −5.56799639350381459035735972316, −5.56247320938621851542079066865, −4.67916414015477345829011530365, −4.62895493864919932696958207991, −3.69936007845062307469290264697, −3.59694594482980948871667656611, −2.85505538044914591947158847584, −2.74571579433716529157191727067, −1.76104732213059710084928950594, −1.67246948067143710738182084046, −0.903442774851921596863683142611, −0.845779668306668128899133781579, 0.845779668306668128899133781579, 0.903442774851921596863683142611, 1.67246948067143710738182084046, 1.76104732213059710084928950594, 2.74571579433716529157191727067, 2.85505538044914591947158847584, 3.59694594482980948871667656611, 3.69936007845062307469290264697, 4.62895493864919932696958207991, 4.67916414015477345829011530365, 5.56247320938621851542079066865, 5.56799639350381459035735972316, 6.49290243910954680031437989975, 6.50217067955517898685784948115, 6.65940982850292086499004836608, 6.72180655486050609785172066075, 7.76982738325878983611731111009, 7.85159449087238944330398236164, 8.438068326580670186621370028747, 8.465464335452150337620407229382

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