Properties

Label 4-22e2-1.1-c9e2-0-0
Degree $4$
Conductor $484$
Sign $1$
Analytic cond. $128.386$
Root an. cond. $3.36612$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·2-s − 21·3-s + 768·4-s − 521·5-s + 672·6-s − 7.49e3·7-s − 1.63e4·8-s − 2.10e4·9-s + 1.66e4·10-s + 2.92e4·11-s − 1.61e4·12-s + 1.50e5·13-s + 2.39e5·14-s + 1.09e4·15-s + 3.27e5·16-s + 6.90e5·17-s + 6.73e5·18-s + 5.11e5·19-s − 4.00e5·20-s + 1.57e5·21-s − 9.37e5·22-s + 8.74e5·23-s + 3.44e5·24-s − 1.61e6·25-s − 4.81e6·26-s + 4.79e5·27-s − 5.75e6·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.149·3-s + 3/2·4-s − 0.372·5-s + 0.211·6-s − 1.17·7-s − 1.41·8-s − 1.06·9-s + 0.527·10-s + 0.603·11-s − 0.224·12-s + 1.45·13-s + 1.66·14-s + 0.0558·15-s + 5/4·16-s + 2.00·17-s + 1.51·18-s + 0.899·19-s − 0.559·20-s + 0.176·21-s − 0.852·22-s + 0.651·23-s + 0.211·24-s − 0.825·25-s − 2.06·26-s + 0.173·27-s − 1.76·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(128.386\)
Root analytic conductor: \(3.36612\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 484,\ (\ :9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.7042232530\)
\(L(\frac12)\) \(\approx\) \(0.7042232530\)
\(L(\frac{11}{2})\) 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$C_1$ \( ( 1 + p^{4} T )^{2} \)
11$C_1$ \( ( 1 - p^{4} T )^{2} \)
good3$D_{4}$ \( 1 + 7 p T + 2386 p^{2} T^{2} + 7 p^{10} T^{3} + p^{18} T^{4} \)
5$D_{4}$ \( 1 + 521 T + 376592 p T^{2} + 521 p^{9} T^{3} + p^{18} T^{4} \)
7$D_{4}$ \( 1 + 1070 p T + 10075602 p T^{2} + 1070 p^{10} T^{3} + p^{18} T^{4} \)
13$D_{4}$ \( 1 - 150314 T + 26804211170 T^{2} - 150314 p^{9} T^{3} + p^{18} T^{4} \)
17$D_{4}$ \( 1 - 40616 p T + 347226119086 T^{2} - 40616 p^{10} T^{3} + p^{18} T^{4} \)
19$D_{4}$ \( 1 - 511212 T + 409533659590 T^{2} - 511212 p^{9} T^{3} + p^{18} T^{4} \)
23$D_{4}$ \( 1 - 874751 T + 3350341409974 T^{2} - 874751 p^{9} T^{3} + p^{18} T^{4} \)
29$D_{4}$ \( 1 + 2951058 T + 17050144887898 T^{2} + 2951058 p^{9} T^{3} + p^{18} T^{4} \)
31$D_{4}$ \( 1 + 5818705 T + 31374188947838 T^{2} + 5818705 p^{9} T^{3} + p^{18} T^{4} \)
37$D_{4}$ \( 1 - 2658905 T + 225492089266940 T^{2} - 2658905 p^{9} T^{3} + p^{18} T^{4} \)
41$D_{4}$ \( 1 - 13427994 T + 688202282720050 T^{2} - 13427994 p^{9} T^{3} + p^{18} T^{4} \)
43$D_{4}$ \( 1 + 17820762 T + 804875927328526 T^{2} + 17820762 p^{9} T^{3} + p^{18} T^{4} \)
47$D_{4}$ \( 1 - 56044104 T + 2070094956889822 T^{2} - 56044104 p^{9} T^{3} + p^{18} T^{4} \)
53$D_{4}$ \( 1 - 96842752 T + 7465684657276486 T^{2} - 96842752 p^{9} T^{3} + p^{18} T^{4} \)
59$D_{4}$ \( 1 + 119136183 T + 5810749115433730 T^{2} + 119136183 p^{9} T^{3} + p^{18} T^{4} \)
61$D_{4}$ \( 1 + 1482366 p T - 2562416206425278 T^{2} + 1482366 p^{10} T^{3} + p^{18} T^{4} \)
67$D_{4}$ \( 1 + 295944891 T + 69921832215701962 T^{2} + 295944891 p^{9} T^{3} + p^{18} T^{4} \)
71$D_{4}$ \( 1 + 322953267 T + 117737224590230422 T^{2} + 322953267 p^{9} T^{3} + p^{18} T^{4} \)
73$D_{4}$ \( 1 + 255975514 T + 133557087434677034 T^{2} + 255975514 p^{9} T^{3} + p^{18} T^{4} \)
79$D_{4}$ \( 1 + 889658 T + 151981069000373118 T^{2} + 889658 p^{9} T^{3} + p^{18} T^{4} \)
83$D_{4}$ \( 1 + 277699042 T + 33227646296175022 T^{2} + 277699042 p^{9} T^{3} + p^{18} T^{4} \)
89$D_{4}$ \( 1 - 1363672217 T + 1117996341977241880 T^{2} - 1363672217 p^{9} T^{3} + p^{18} T^{4} \)
97$D_{4}$ \( 1 - 1398434043 T + 1658486726134145236 T^{2} - 1398434043 p^{9} T^{3} + p^{18} T^{4} \)
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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

−16.33866143523375467600717014945, −15.98862612993314410639477841416, −15.11972734325486814328519758322, −14.46831021001282356892790031732, −13.63124310535358002612806001329, −12.79303822713282876366190642448, −11.75975307575129699898920843945, −11.68806574079120524385995203303, −10.68515759156517896957359790126, −10.07948358614585014368763106035, −9.039668237516440359554431463109, −9.020640629053368125138312110931, −7.77038861335200721854753370759, −7.30359298897399984863097056950, −5.95245196720830927398293647809, −5.85929282597650854684097094746, −3.56023544781123300836964739338, −3.10734395935336692249976399591, −1.41581467294580684583261045009, −0.51678432464801416817963132739, 0.51678432464801416817963132739, 1.41581467294580684583261045009, 3.10734395935336692249976399591, 3.56023544781123300836964739338, 5.85929282597650854684097094746, 5.95245196720830927398293647809, 7.30359298897399984863097056950, 7.77038861335200721854753370759, 9.020640629053368125138312110931, 9.039668237516440359554431463109, 10.07948358614585014368763106035, 10.68515759156517896957359790126, 11.68806574079120524385995203303, 11.75975307575129699898920843945, 12.79303822713282876366190642448, 13.63124310535358002612806001329, 14.46831021001282356892790031732, 15.11972734325486814328519758322, 15.98862612993314410639477841416, 16.33866143523375467600717014945

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