Properties

Label 4-3822e2-1.1-c1e2-0-11
Degree $4$
Conductor $14607684$
Sign $1$
Analytic cond. $931.398$
Root an. cond. $5.52438$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 4·8-s + 3·9-s − 4·10-s + 2·11-s − 6·12-s − 2·13-s − 4·15-s + 5·16-s + 2·17-s − 6·18-s + 2·19-s + 6·20-s − 4·22-s − 6·23-s + 8·24-s − 7·25-s + 4·26-s − 4·27-s − 6·29-s + 8·30-s − 6·32-s − 4·33-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.41·8-s + 9-s − 1.26·10-s + 0.603·11-s − 1.73·12-s − 0.554·13-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s + 0.458·19-s + 1.34·20-s − 0.852·22-s − 1.25·23-s + 1.63·24-s − 7/5·25-s + 0.784·26-s − 0.769·27-s − 1.11·29-s + 1.46·30-s − 1.06·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14607684\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(931.398\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 14607684,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 169 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 132 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 96 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 18 T + 3 p T^{2} + 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 176 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 180 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.046255284652506278349440589318, −7.979639116509307860586301761804, −7.54187931310204115467991397435, −7.47717444341696420356416605910, −6.66553271887956183356488485778, −6.54777418950223972095115932925, −6.07727344296996170865904915860, −5.95235842153120318319577140052, −5.33735176862639062986958347826, −5.32022954134583121949417701250, −4.38211573493227544096717014405, −4.31248723275319744200428680540, −3.39552933366423361009654806261, −3.30474825443667136373169267406, −2.23975415598967972535426682463, −2.15614595230913994198923739527, −1.36230987254613895198619212043, −1.31397148507771552339153423959, 0, 0, 1.31397148507771552339153423959, 1.36230987254613895198619212043, 2.15614595230913994198923739527, 2.23975415598967972535426682463, 3.30474825443667136373169267406, 3.39552933366423361009654806261, 4.31248723275319744200428680540, 4.38211573493227544096717014405, 5.32022954134583121949417701250, 5.33735176862639062986958347826, 5.95235842153120318319577140052, 6.07727344296996170865904915860, 6.54777418950223972095115932925, 6.66553271887956183356488485778, 7.47717444341696420356416605910, 7.54187931310204115467991397435, 7.979639116509307860586301761804, 8.046255284652506278349440589318

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