# Properties

 Label 8-2352e4-1.1-c1e4-0-6 Degree $8$ Conductor $3.060\times 10^{13}$ Sign $1$ Analytic cond. $124410.$ Root an. cond. $4.33368$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 5·9-s + 16·13-s − 2·25-s − 4·37-s − 12·61-s − 28·73-s + 16·81-s − 32·97-s + 20·109-s + 80·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
 L(s)  = 1 + 5/3·9-s + 4.43·13-s − 2/5·25-s − 0.657·37-s − 1.53·61-s − 3.27·73-s + 16/9·81-s − 3.24·97-s + 1.91·109-s + 7.39·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{4} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$124410.$$ Root analytic conductor: $$4.33368$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2352} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.018619873$$ $$L(\frac12)$$ $$\approx$$ $$3.018619873$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
7 $$1$$
good5$C_2$ $$( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2}$$
11$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
17$C_2^2$ $$( 1 - 23 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 11 T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^2$ $$( 1 + 35 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 14 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 53 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2$ $$( 1 + T + p T^{2} )^{4}$$
41$C_2^2$ $$( 1 - 38 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 82 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 5 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 95 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 107 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 + 3 T + p T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 53 T^{2} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 34 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2$ $$( 1 + 7 T + p T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 77 T^{2} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 167 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−6.38138251939944851980219349958, −6.11538298672732416983799848957, −5.84259545461250534936219790857, −5.77399626508176851761961432814, −5.76940372992010244081191866557, −5.62983841764269168220642875609, −4.95205519805719681484466041083, −4.77913634018938331496041938545, −4.68696232001401590122359276730, −4.57934455026507353882062436082, −4.07675278987242159295344336847, −3.97147758249226579107430812738, −3.80365606417093725535866851125, −3.62892121347177518730254026326, −3.48877681944885302994491703911, −3.17337151091118136168354972091, −2.96126512595993418245161730918, −2.46671361603398062678100262791, −2.37320654341794681629595739375, −1.83974472691206594939570096136, −1.55834919507136475650581347254, −1.22256879844287843679056171104, −1.20762295331120068890166440811, −1.19423749366218509530302355984, −0.24510044715582581986988388617, 0.24510044715582581986988388617, 1.19423749366218509530302355984, 1.20762295331120068890166440811, 1.22256879844287843679056171104, 1.55834919507136475650581347254, 1.83974472691206594939570096136, 2.37320654341794681629595739375, 2.46671361603398062678100262791, 2.96126512595993418245161730918, 3.17337151091118136168354972091, 3.48877681944885302994491703911, 3.62892121347177518730254026326, 3.80365606417093725535866851125, 3.97147758249226579107430812738, 4.07675278987242159295344336847, 4.57934455026507353882062436082, 4.68696232001401590122359276730, 4.77913634018938331496041938545, 4.95205519805719681484466041083, 5.62983841764269168220642875609, 5.76940372992010244081191866557, 5.77399626508176851761961432814, 5.84259545461250534936219790857, 6.11538298672732416983799848957, 6.38138251939944851980219349958