# Properties

 Label 4-950e2-1.1-c5e2-0-1 Degree $4$ Conductor $902500$ Sign $1$ Analytic cond. $23214.9$ Root an. cond. $12.3436$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·2-s − 3·3-s + 48·4-s − 24·6-s − 114·7-s + 256·8-s − 119·9-s + 661·11-s − 144·12-s − 1.61e3·13-s − 912·14-s + 1.28e3·16-s − 64·17-s − 952·18-s + 722·19-s + 342·21-s + 5.28e3·22-s + 3.18e3·23-s − 768·24-s − 1.29e4·26-s + 12·27-s − 5.47e3·28-s − 2.48e3·29-s − 1.18e3·31-s + 6.14e3·32-s − 1.98e3·33-s − 512·34-s + ⋯
 L(s)  = 1 + 1.41·2-s − 0.192·3-s + 3/2·4-s − 0.272·6-s − 0.879·7-s + 1.41·8-s − 0.489·9-s + 1.64·11-s − 0.288·12-s − 2.64·13-s − 1.24·14-s + 5/4·16-s − 0.0537·17-s − 0.692·18-s + 0.458·19-s + 0.169·21-s + 2.32·22-s + 1.25·23-s − 0.272·24-s − 3.74·26-s + 0.00316·27-s − 1.31·28-s − 0.547·29-s − 0.220·31-s + 1.06·32-s − 0.316·33-s − 0.0759·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$902500$$    =    $$2^{2} \cdot 5^{4} \cdot 19^{2}$$ Sign: $1$ Analytic conductor: $$23214.9$$ Root analytic conductor: $$12.3436$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{950} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 902500,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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$C_1$ $$( 1 - p^{2} T )^{2}$$
5 $$1$$
19$C_1$ $$( 1 - p^{2} T )^{2}$$
good3$D_{4}$ $$1 + p T + 128 T^{2} + p^{6} T^{3} + p^{10} T^{4}$$
7$D_{4}$ $$1 + 114 T + 31099 T^{2} + 114 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 - 661 T + 430972 T^{2} - 661 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 + 1613 T + 1384022 T^{2} + 1613 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 + 64 T + 2804713 T^{2} + 64 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 - 3185 T + 14543782 T^{2} - 3185 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 2481 T + 6815332 T^{2} + 2481 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 1180 T + 21633278 T^{2} + 1180 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 + 10488 T + 155529814 T^{2} + 10488 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 16630 T + 170295586 T^{2} - 16630 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 11303 T + 297510638 T^{2} + 11303 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 - 12155 T + 47754214 T^{2} - 12155 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 + 20585 T + 812203882 T^{2} + 20585 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 + 78581 T + 2971672216 T^{2} + 78581 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 - 43621 T + 1919695356 T^{2} - 43621 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 + 7805 T - 748756690 T^{2} + 7805 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 + 62488 T + 4005427642 T^{2} + 62488 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 + 16218 T + 1004140843 T^{2} + 16218 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 - 67122 T + 7273984870 T^{2} - 67122 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 10714 T + 5751433246 T^{2} - 10714 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 - 128188 T + 8689285330 T^{2} - 128188 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 + 178558 T + 22668743394 T^{2} + 178558 p^{5} T^{3} + p^{10} T^{4}$$
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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

−9.129434144609116193537200684772, −9.030320106263371284530200187759, −7.926825121170063782183372099451, −7.69905186000330746030367434424, −7.14630541703623830423773226349, −6.89283237113185984173114475938, −6.41566966873357572970608230544, −6.18982834416182642654956277109, −5.32681871822788878088288475727, −5.29162062857896153583668196511, −4.71336109804064401555062674096, −4.28163452685650426557895937720, −3.72595559469754306097378865617, −3.20689990266173227905758790300, −2.83144276114041174353568395192, −2.39867039197355751635816936463, −1.62859656729032328239270440338, −1.16302896072277631483145537948, 0, 0, 1.16302896072277631483145537948, 1.62859656729032328239270440338, 2.39867039197355751635816936463, 2.83144276114041174353568395192, 3.20689990266173227905758790300, 3.72595559469754306097378865617, 4.28163452685650426557895937720, 4.71336109804064401555062674096, 5.29162062857896153583668196511, 5.32681871822788878088288475727, 6.18982834416182642654956277109, 6.41566966873357572970608230544, 6.89283237113185984173114475938, 7.14630541703623830423773226349, 7.69905186000330746030367434424, 7.926825121170063782183372099451, 9.030320106263371284530200187759, 9.129434144609116193537200684772