Properties

Degree $4$
Conductor $1444$
Sign $1$
Motivic weight $7$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s − 69·3-s + 192·4-s + 155·5-s − 1.10e3·6-s − 2.23e3·7-s + 2.04e3·8-s + 621·9-s + 2.48e3·10-s − 3.29e3·11-s − 1.32e4·12-s − 1.34e4·13-s − 3.58e4·14-s − 1.06e4·15-s + 2.04e4·16-s − 3.22e4·17-s + 9.93e3·18-s − 1.37e4·19-s + 2.97e4·20-s + 1.54e5·21-s − 5.27e4·22-s − 8.25e4·23-s − 1.41e5·24-s + 1.38e4·25-s − 2.14e5·26-s + 9.19e4·27-s − 4.29e5·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.47·3-s + 3/2·4-s + 0.554·5-s − 2.08·6-s − 2.46·7-s + 1.41·8-s + 0.283·9-s + 0.784·10-s − 0.746·11-s − 2.21·12-s − 1.69·13-s − 3.48·14-s − 0.818·15-s + 5/4·16-s − 1.59·17-s + 0.401·18-s − 0.458·19-s + 0.831·20-s + 3.63·21-s − 1.05·22-s − 1.41·23-s − 2.08·24-s + 0.177·25-s − 2.39·26-s + 0.898·27-s − 3.69·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(7\)
Character: induced by $\chi_{38} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1444,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 - p^{3} T )^{2} \)
19$C_1$ \( ( 1 + p^{3} T )^{2} \)
good3$D_{4}$ \( 1 + 23 p T + 460 p^{2} T^{2} + 23 p^{8} T^{3} + p^{14} T^{4} \)
5$D_{4}$ \( 1 - 31 p T + 10178 T^{2} - 31 p^{8} T^{3} + p^{14} T^{4} \)
7$D_{4}$ \( 1 + 2238 T + 2775179 T^{2} + 2238 p^{7} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 + 3295 T + 38080340 T^{2} + 3295 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 + 13427 T + 138664758 T^{2} + 13427 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 + 32256 T + 1073810905 T^{2} + 32256 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 + 82525 T + 5722043942 T^{2} + 82525 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 + 12749 T + 39176144 p^{2} T^{2} + 12749 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 - 258944 T + 64961539758 T^{2} - 258944 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 + 149260 T + 183707435838 T^{2} + 149260 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 - 339130 T - 4364095006 T^{2} - 339130 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 + 83869 T + 6409629042 p T^{2} + 83869 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 - 1471025 T + 1488141383726 T^{2} - 1471025 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 + 945643 T + 2428814565902 T^{2} + 945643 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 + 969009 T + 3139777837024 T^{2} + 969009 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 + 1506755 T + 5925806441724 T^{2} + 1506755 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 + 1848219 T + 11014380276458 T^{2} + 1848219 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 + 3417184 T + 7686276501674 T^{2} + 3417184 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 + 2499822 T + 17574764549507 T^{2} + 2499822 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 2636926 T + 33245149452510 T^{2} - 2636926 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 + 10059354 T + 77358130536910 T^{2} + 10059354 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 + 3506160 T + 657141629522 p T^{2} + 3506160 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 - 60758 p T + 73244272821042 T^{2} - 60758 p^{8} T^{3} + p^{14} T^{4} \)
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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

−14.21830421122488668237391438244, −13.79386770200144625299144922712, −12.93745289358173204480144830356, −12.89822855690209524814218253750, −12.08202323532062773319172066580, −11.86790942981865475035807348446, −10.85855714937440245662825922801, −10.28772306409231108133482774754, −9.886381209645362185797124064715, −8.979835451662045915112157503590, −7.49298622933437872789889365905, −6.71448189102433090559053016916, −6.04600433220233326194543298214, −6.02667985244290819787978193283, −4.97474178816789664448980853352, −4.22970940818352579430758329214, −2.87694526840743978921992319847, −2.42938874675819712221643987754, 0, 0, 2.42938874675819712221643987754, 2.87694526840743978921992319847, 4.22970940818352579430758329214, 4.97474178816789664448980853352, 6.02667985244290819787978193283, 6.04600433220233326194543298214, 6.71448189102433090559053016916, 7.49298622933437872789889365905, 8.979835451662045915112157503590, 9.886381209645362185797124064715, 10.28772306409231108133482774754, 10.85855714937440245662825922801, 11.86790942981865475035807348446, 12.08202323532062773319172066580, 12.89822855690209524814218253750, 12.93745289358173204480144830356, 13.79386770200144625299144922712, 14.21830421122488668237391438244

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