Properties

Label 4-1216e2-1.1-c3e2-0-9
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $5147.53$
Root an. cond. $8.47032$
Motivic weight $3$
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  − 3-s − 10·5-s + 57·7-s − 9·9-s − 10·11-s − 13·13-s + 10·15-s − 51·17-s + 38·19-s − 57·21-s − 155·23-s + 2·25-s − 8·27-s + 79·29-s − 16·31-s + 10·33-s − 570·35-s − 380·37-s + 13·39-s − 790·41-s − 296·43-s + 90·45-s − 200·47-s + 1.79e3·49-s + 51·51-s − 397·53-s + 100·55-s + ⋯
L(s)  = 1  − 0.192·3-s − 0.894·5-s + 3.07·7-s − 1/3·9-s − 0.274·11-s − 0.277·13-s + 0.172·15-s − 0.727·17-s + 0.458·19-s − 0.592·21-s − 1.40·23-s + 0.0159·25-s − 0.0570·27-s + 0.505·29-s − 0.0926·31-s + 0.0527·33-s − 2.75·35-s − 1.68·37-s + 0.0533·39-s − 3.00·41-s − 1.04·43-s + 0.298·45-s − 0.620·47-s + 5.23·49-s + 0.140·51-s − 1.02·53-s + 0.245·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5147.53\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1478656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - p T )^{2} \)
good3$D_{4}$ \( 1 + T + 10 T^{2} + p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 + 2 p T + 98 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 57 T + 1454 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 10 T + 2510 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + p T + 2268 T^{2} + p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 3 p T + 520 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 155 T + 994 p T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 79 T + 13124 T^{2} - 79 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 16 T + 48318 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 380 T + 126078 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 790 T + 292274 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 296 T + 78966 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 200 T + 146846 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 397 T + 333572 T^{2} + 397 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 201 T + 197794 T^{2} + 201 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 680 T + 483894 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 939 T + 740138 T^{2} - 939 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 406 T + 735614 T^{2} - 406 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 123 T + 781772 T^{2} - 123 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 106 T + 840030 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 2226 T + 2380750 T^{2} + 2226 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 870 T + 1594738 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1864 T + 2438382 T^{2} + 1864 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.833066034356056374839418273832, −8.508431743765844541609911327994, −8.217693162887477481199095857237, −8.139547933184006834108158182348, −7.71589192265001690673397306665, −7.10834927207361940618908678172, −6.84681336586784142540283497290, −6.29980641027924536595657934051, −5.32787094042703285683751015552, −5.24098438959926866780803208443, −5.07900773354691242305077481380, −4.43507876829310594203645225167, −4.00953476725385665924750331071, −3.61099934771943815120791393758, −2.76956700242838157787831580348, −2.12941343105458902816175541859, −1.52691469877841808685155278207, −1.42035075968957024892226914283, 0, 0, 1.42035075968957024892226914283, 1.52691469877841808685155278207, 2.12941343105458902816175541859, 2.76956700242838157787831580348, 3.61099934771943815120791393758, 4.00953476725385665924750331071, 4.43507876829310594203645225167, 5.07900773354691242305077481380, 5.24098438959926866780803208443, 5.32787094042703285683751015552, 6.29980641027924536595657934051, 6.84681336586784142540283497290, 7.10834927207361940618908678172, 7.71589192265001690673397306665, 8.139547933184006834108158182348, 8.217693162887477481199095857237, 8.508431743765844541609911327994, 8.833066034356056374839418273832

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