Properties

Label 4-99e2-1.1-c3e2-0-3
Degree $4$
Conductor $9801$
Sign $1$
Analytic cond. $34.1194$
Root an. cond. $2.41685$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 9·4-s + 14·5-s + 24·7-s − 25·8-s − 14·10-s + 22·11-s + 30·13-s − 24·14-s + 41·16-s − 106·17-s + 50·19-s + 126·20-s − 22·22-s − 134·23-s − 6·25-s − 30·26-s + 216·28-s + 198·29-s + 360·31-s − 249·32-s + 106·34-s + 336·35-s − 328·37-s − 50·38-s − 350·40-s + 782·41-s + ⋯
L(s)  = 1  − 0.353·2-s + 9/8·4-s + 1.25·5-s + 1.29·7-s − 1.10·8-s − 0.442·10-s + 0.603·11-s + 0.640·13-s − 0.458·14-s + 0.640·16-s − 1.51·17-s + 0.603·19-s + 1.40·20-s − 0.213·22-s − 1.21·23-s − 0.0479·25-s − 0.226·26-s + 1.45·28-s + 1.26·29-s + 2.08·31-s − 1.37·32-s + 0.534·34-s + 1.62·35-s − 1.45·37-s − 0.213·38-s − 1.38·40-s + 2.97·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9801\)    =    \(3^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(34.1194\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9801,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.129349594\)
\(L(\frac12)\) \(\approx\) \(3.129349594\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
11$C_1$ \( ( 1 - p T )^{2} \)
good2$D_{4}$ \( 1 + T - p^{3} T^{2} + p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 14 T + 202 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 24 T + 442 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 30 T + 4522 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 106 T + 7882 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 50 T + 14246 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 134 T + 26398 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 198 T + 57706 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 360 T + 90430 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 328 T + 62630 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 782 T + 285970 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 386 T + 179870 T^{2} - 386 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 266 T + 92542 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 522 T + 295162 T^{2} - 522 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 172 T + 175654 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 778 T + 577250 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 776 T + 528582 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 630 T + 744334 T^{2} + 630 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1296 T + 1178926 T^{2} - 1296 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 652 T + 589506 T^{2} - 652 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 324 T + 579670 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 756 T + 1427110 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 452 T + 982470 T^{2} + 452 p^{3} T^{3} + p^{6} T^{4} \)
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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

−13.83131033657998373805596284139, −13.36357204796690508666455791442, −12.33042169421354542700786159364, −11.98087991122259990076142989170, −11.59710886240666788066540185186, −10.81260124084478207474265939611, −10.74062031064241600045840746357, −9.886763094665814527666032787237, −9.278275354698618631066677104851, −8.831949865299711343713697297888, −8.177995380451782918339950309412, −7.56871344248201158799465989825, −6.70808927952813847529213459907, −6.14865234013846619593981185274, −5.93752445879423514060989834098, −4.85682512301971164524350743589, −4.05716165334245126444049618862, −2.64075537766779952728152551290, −2.12076745903448379322719996426, −1.20457236984055875159994227351, 1.20457236984055875159994227351, 2.12076745903448379322719996426, 2.64075537766779952728152551290, 4.05716165334245126444049618862, 4.85682512301971164524350743589, 5.93752445879423514060989834098, 6.14865234013846619593981185274, 6.70808927952813847529213459907, 7.56871344248201158799465989825, 8.177995380451782918339950309412, 8.831949865299711343713697297888, 9.278275354698618631066677104851, 9.886763094665814527666032787237, 10.74062031064241600045840746357, 10.81260124084478207474265939611, 11.59710886240666788066540185186, 11.98087991122259990076142989170, 12.33042169421354542700786159364, 13.36357204796690508666455791442, 13.83131033657998373805596284139

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