Properties

Label 8-2352e4-1.1-c1e4-0-8
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·9-s − 14·25-s − 8·37-s + 32·43-s − 8·67-s + 4·79-s + 27·81-s − 8·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2·9-s − 2.79·25-s − 1.31·37-s + 4.87·43-s − 0.977·67-s + 0.450·79-s + 3·81-s − 0.766·109-s + 2.36·121-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 + 2.15·169-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 + 0.0708·199-s + 0.0688·211-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

\(L(1)\) \(\approx\) \(2.551247407\)
\(L(\frac12)\) \(\approx\) \(2.551247407\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 115 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \)
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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

−6.21444859448938838738504863741, −6.10588429350330518068836053848, −5.90188329770662818536486429691, −5.86932208953278291030543797542, −5.75984353372290186025470334324, −5.31265487807513521547279458515, −5.00710322664852881674739958525, −4.87022266051484069253859272918, −4.70107253505694558494188322205, −4.54879398478123741393420424227, −4.02188095920763007795072079486, −3.93323059058424108423462293932, −3.92544045986517374894828627205, −3.81250921988427487955264754089, −3.47093354185114343234046153187, −3.10682422738816522120548439032, −2.73492008122399628671480802841, −2.39088190902391047253364628434, −2.24193870786644059789294622175, −2.23900723230052960681721741508, −1.54463619521150421344988691508, −1.49691203357612505158670658876, −1.21047555105489621528029176321, −0.77714708223567794295784758873, −0.26934485923986568388027407714, 0.26934485923986568388027407714, 0.77714708223567794295784758873, 1.21047555105489621528029176321, 1.49691203357612505158670658876, 1.54463619521150421344988691508, 2.23900723230052960681721741508, 2.24193870786644059789294622175, 2.39088190902391047253364628434, 2.73492008122399628671480802841, 3.10682422738816522120548439032, 3.47093354185114343234046153187, 3.81250921988427487955264754089, 3.92544045986517374894828627205, 3.93323059058424108423462293932, 4.02188095920763007795072079486, 4.54879398478123741393420424227, 4.70107253505694558494188322205, 4.87022266051484069253859272918, 5.00710322664852881674739958525, 5.31265487807513521547279458515, 5.75984353372290186025470334324, 5.86932208953278291030543797542, 5.90188329770662818536486429691, 6.10588429350330518068836053848, 6.21444859448938838738504863741

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