Properties

Label 16-378e8-1.1-c1e8-0-0
Degree $16$
Conductor $4.168\times 10^{20}$
Sign $1$
Analytic cond. $6888.92$
Root an. cond. $1.73733$
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  + 2·4-s − 8·7-s + 16-s + 8·25-s − 16·28-s − 12·31-s − 4·37-s + 56·43-s + 12·49-s + 12·61-s − 2·64-s − 20·67-s − 72·73-s − 20·79-s + 16·100-s + 60·103-s + 20·109-s − 8·112-s − 8·121-s − 24·124-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 4-s − 3.02·7-s + 1/4·16-s + 8/5·25-s − 3.02·28-s − 2.15·31-s − 0.657·37-s + 8.53·43-s + 12/7·49-s + 1.53·61-s − 1/4·64-s − 2.44·67-s − 8.42·73-s − 2.25·79-s + 8/5·100-s + 5.91·103-s + 1.91·109-s − 0.755·112-s − 0.727·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{24} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6888.92\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{378} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.04308282595\)
\(L(\frac12)\) \(\approx\) \(0.04308282595\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{2} \)
3 \( 1 \)
7 \( ( 1 + 2 T + p T^{2} )^{4} \)
good5 \( ( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 4 T^{2} - 105 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 34 T^{2} + 555 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 14 T^{2} - 165 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 8 T^{2} - 894 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 66 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 34 p T^{5} - 53 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 7 T + p T^{2} )^{8} \)
47 \( 1 + 40 T^{2} - 626 T^{4} - 87680 T^{6} - 3459005 T^{8} - 87680 p^{2} T^{10} - 626 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 4 T^{2} + 4762 T^{4} + 41456 T^{6} + 14589091 T^{8} + 41456 p^{2} T^{10} + 4762 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 112 T^{2} + 9063 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 6 T + 131 T^{2} - 714 T^{3} + 11172 T^{4} - 714 p T^{5} + 131 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 10 T + 13 T^{2} - 470 T^{3} - 3620 T^{4} - 470 p T^{5} + 13 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 79107 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 70 p T^{5} - 65 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 68 T^{2} + 4566 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( 1 - 32 T^{2} + 8254 T^{4} + 738304 T^{6} - 18580445 T^{8} + 738304 p^{2} T^{10} + 8254 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 - 334 T^{2} + 46419 T^{4} - 334 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.04028860507216640092781304137, −4.79968519774219734071919844863, −4.65677218590670936953521271234, −4.61163258924878344494799853041, −4.39119045265655951491120401214, −4.33089915590753635755573465160, −4.18672145210840021439404454344, −4.02184673082405158798424180172, −3.87550330384995738353840016764, −3.49926994953112822885726444830, −3.40981879657180536897900609000, −3.33064832592204535138309223376, −3.22905340678241951066975829929, −3.04042345292778083799062935256, −2.81855434303915579719796488249, −2.70371834603069729774990166611, −2.62373335655672969775321743652, −2.32297116541641727875467440390, −2.17357927810151831613887630906, −2.04013866467793594819661442217, −1.56594491512459227818753684601, −1.37216278210050611772860377394, −1.06959992285701514232957566219, −0.824009368639995085470008281421, −0.04669036181114440510663954777, 0.04669036181114440510663954777, 0.824009368639995085470008281421, 1.06959992285701514232957566219, 1.37216278210050611772860377394, 1.56594491512459227818753684601, 2.04013866467793594819661442217, 2.17357927810151831613887630906, 2.32297116541641727875467440390, 2.62373335655672969775321743652, 2.70371834603069729774990166611, 2.81855434303915579719796488249, 3.04042345292778083799062935256, 3.22905340678241951066975829929, 3.33064832592204535138309223376, 3.40981879657180536897900609000, 3.49926994953112822885726444830, 3.87550330384995738353840016764, 4.02184673082405158798424180172, 4.18672145210840021439404454344, 4.33089915590753635755573465160, 4.39119045265655951491120401214, 4.61163258924878344494799853041, 4.65677218590670936953521271234, 4.79968519774219734071919844863, 5.04028860507216640092781304137

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

Plot not available for L-functions of degree greater than 10.