Properties

Label 4-4598e2-1.1-c1e2-0-14
Degree $4$
Conductor $21141604$
Sign $1$
Analytic cond. $1348.00$
Root an. cond. $6.05930$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 3·4-s − 5-s − 2·6-s + 7-s + 4·8-s − 4·9-s − 2·10-s − 3·12-s + 13-s + 2·14-s + 15-s + 5·16-s − 2·17-s − 8·18-s + 2·19-s − 3·20-s − 21-s − 4·23-s − 4·24-s − 8·25-s + 2·26-s + 6·27-s + 3·28-s − 5·29-s + 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 1.41·8-s − 4/3·9-s − 0.632·10-s − 0.866·12-s + 0.277·13-s + 0.534·14-s + 0.258·15-s + 5/4·16-s − 0.485·17-s − 1.88·18-s + 0.458·19-s − 0.670·20-s − 0.218·21-s − 0.834·23-s − 0.816·24-s − 8/5·25-s + 0.392·26-s + 1.15·27-s + 0.566·28-s − 0.928·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21141604\)    =    \(2^{2} \cdot 11^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1348.00\)
Root analytic conductor: \(6.05930\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 21141604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + T + 81 T^{2} + p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 19 T + 221 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 138 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 7 T + 177 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 28 T + 370 T^{2} + 28 p T^{3} + p^{2} T^{4} \)
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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

−8.079669302764907765894245251210, −7.929007849642636159687292584250, −7.12102679703490535698028213871, −7.06641467689381830597031919905, −6.43653090598679040491518179036, −6.28485070753969783161114021682, −5.82217423965232809617317473378, −5.51707681286640136353231277711, −5.13804168187903940118684905143, −5.01350273666736267240843715141, −4.40756861604253388728898886618, −3.91313313259222151379392665912, −3.61046177234654650859244430183, −3.41550729580287004738711400103, −2.66430133731107815385021548385, −2.42482992612592506085229127537, −1.69030125648353964306231340091, −1.42460999369936152160007923338, 0, 0, 1.42460999369936152160007923338, 1.69030125648353964306231340091, 2.42482992612592506085229127537, 2.66430133731107815385021548385, 3.41550729580287004738711400103, 3.61046177234654650859244430183, 3.91313313259222151379392665912, 4.40756861604253388728898886618, 5.01350273666736267240843715141, 5.13804168187903940118684905143, 5.51707681286640136353231277711, 5.82217423965232809617317473378, 6.28485070753969783161114021682, 6.43653090598679040491518179036, 7.06641467689381830597031919905, 7.12102679703490535698028213871, 7.929007849642636159687292584250, 8.079669302764907765894245251210

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