Properties

Label 8-825e4-1.1-c1e4-0-7
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
Motivic weight $1$
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  + 7·16-s − 24·29-s + 16·31-s + 8·61-s − 9·81-s + 48·89-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 7/4·16-s − 4.45·29-s + 2.87·31-s + 1.02·61-s − 81-s + 5.08·89-s − 0.181·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.235977248\)
\(L(\frac12)\) \(\approx\) \(3.235977248\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$C_2^3$ \( 1 - 7 T^{4} + p^{4} T^{8} \)
7$C_2^3$ \( 1 - 94 T^{4} + p^{4} T^{8} \)
13$C_2^3$ \( 1 - 142 T^{4} + p^{4} T^{8} \)
17$C_2^3$ \( 1 - 382 T^{4} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 1054 T^{4} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 - 3214 T^{4} + p^{4} T^{8} \)
47$C_2^3$ \( 1 - 2302 T^{4} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 2254 T^{4} + p^{4} T^{8} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
67$C_2^3$ \( 1 + 5906 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 + 7298 T^{4} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 10414 T^{4} + p^{4} T^{8} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
97$C_2^3$ \( 1 - 18814 T^{4} + p^{4} T^{8} \)
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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

−7.37054315867894742638343854616, −7.14396192436113412820275691100, −6.93668977679232624864165114260, −6.52341874887124592549351828444, −6.46491062830265175332087060097, −6.27769627525307371960362385828, −5.78296719568389731010845163604, −5.69106198843604375285769750219, −5.55690593678711354309153587803, −5.35249808137406888784794079140, −5.07780873691497002046649122971, −4.64232924007728050199956998151, −4.51072760007118927747464899286, −4.14518850466171904284743629190, −3.99890462760827521878461780204, −3.49608128935490271051702355809, −3.42429216389021605845858533824, −3.28767433457833897053219200939, −2.83110845861132922722951454565, −2.44754546425585131364813554200, −2.11547747001274555263406051458, −1.71341675491868712290049234662, −1.58593089089784044143061014768, −0.77848454641705121617388825512, −0.59891573596854514569297922697, 0.59891573596854514569297922697, 0.77848454641705121617388825512, 1.58593089089784044143061014768, 1.71341675491868712290049234662, 2.11547747001274555263406051458, 2.44754546425585131364813554200, 2.83110845861132922722951454565, 3.28767433457833897053219200939, 3.42429216389021605845858533824, 3.49608128935490271051702355809, 3.99890462760827521878461780204, 4.14518850466171904284743629190, 4.51072760007118927747464899286, 4.64232924007728050199956998151, 5.07780873691497002046649122971, 5.35249808137406888784794079140, 5.55690593678711354309153587803, 5.69106198843604375285769750219, 5.78296719568389731010845163604, 6.27769627525307371960362385828, 6.46491062830265175332087060097, 6.52341874887124592549351828444, 6.93668977679232624864165114260, 7.14396192436113412820275691100, 7.37054315867894742638343854616

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