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Properties

Label 1-80e2-6400.1253-r1-0-0
Degree $1$
Conductor $6400$
Sign $0.707 - 0.706i$
Analytic cond. $687.775$
Root an. cond. $687.775$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.693 − 0.720i)3-s + (0.555 + 0.831i)7-s + (−0.0392 − 0.999i)9-s + (0.944 + 0.327i)11-s + (0.0588 − 0.998i)13-s + (−0.996 + 0.0784i)17-s + (0.976 + 0.214i)19-s + (0.984 + 0.175i)21-s + (0.418 − 0.908i)23-s + (−0.747 − 0.664i)27-s + (−0.999 + 0.0196i)29-s + (0.891 − 0.453i)31-s + (0.891 − 0.453i)33-s + (0.505 + 0.862i)37-s + (−0.678 − 0.734i)39-s + ⋯
L(s)  = 1  + (0.693 − 0.720i)3-s + (0.555 + 0.831i)7-s + (−0.0392 − 0.999i)9-s + (0.944 + 0.327i)11-s + (0.0588 − 0.998i)13-s + (−0.996 + 0.0784i)17-s + (0.976 + 0.214i)19-s + (0.984 + 0.175i)21-s + (0.418 − 0.908i)23-s + (−0.747 − 0.664i)27-s + (−0.999 + 0.0196i)29-s + (0.891 − 0.453i)31-s + (0.891 − 0.453i)33-s + (0.505 + 0.862i)37-s + (−0.678 − 0.734i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $0.707 - 0.706i$
Analytic conductor: \(687.775\)
Root analytic conductor: \(687.775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6400} (1253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6400,\ (1:\ ),\ 0.707 - 0.706i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.890484848 - 1.609254646i\)
\(L(\frac12)\) \(\approx\) \(3.890484848 - 1.609254646i\)
\(L(1)\) \(\approx\) \(1.565164058 - 0.3473400445i\)
\(L(1)\) \(\approx\) \(1.565164058 - 0.3473400445i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.693 - 0.720i)T \)
7 \( 1 + (0.555 + 0.831i)T \)
11 \( 1 + (0.944 + 0.327i)T \)
13 \( 1 + (0.0588 - 0.998i)T \)
17 \( 1 + (-0.996 + 0.0784i)T \)
19 \( 1 + (0.976 + 0.214i)T \)
23 \( 1 + (0.418 - 0.908i)T \)
29 \( 1 + (-0.999 + 0.0196i)T \)
31 \( 1 + (0.891 - 0.453i)T \)
37 \( 1 + (0.505 + 0.862i)T \)
41 \( 1 + (0.734 + 0.678i)T \)
43 \( 1 + (-0.471 + 0.881i)T \)
47 \( 1 + (0.649 + 0.760i)T \)
53 \( 1 + (-0.571 - 0.820i)T \)
59 \( 1 + (0.252 - 0.967i)T \)
61 \( 1 + (0.137 - 0.990i)T \)
67 \( 1 + (0.175 + 0.984i)T \)
71 \( 1 + (0.619 - 0.785i)T \)
73 \( 1 + (-0.938 + 0.346i)T \)
79 \( 1 + (0.996 + 0.0784i)T \)
83 \( 1 + (0.976 + 0.214i)T \)
89 \( 1 + (0.678 + 0.734i)T \)
97 \( 1 + (-0.891 - 0.453i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.39398349532300484981752177271, −16.7670137884235952255168482728, −16.26510117973908434515411595915, −15.50666273622800121684249556335, −14.89717572925020000820747172760, −14.18219382118491896990730342470, −13.69624726062598459057659734886, −13.40218091559191884333968747485, −12.12536152088857907784899301172, −11.37524844067880137986819471978, −11.032708282671970830328386454731, −10.245304657237462279324742112730, −9.347541699364687759300300378809, −9.08680461079712285528524785763, −8.37922018126214166217025879756, −7.28620123848135026169503767799, −7.20223093549236825856840733312, −6.03925500795362013353944395769, −5.160204700441841836230059891622, −4.420472490790056977749756474737, −3.92811405161114086531982274495, −3.33451673183438713452728780266, −2.27560904391736151414676781892, −1.59611895483090212699289163708, −0.705298673157198375087515510409, 0.643824217487586994477586366509, 1.34021125550241869583017331945, 2.15354211469807140476889968565, 2.7770772910081067558598648572, 3.50951563847412682510475450441, 4.44773079838489371495434474866, 5.15875595084853147242768946676, 6.20367808975691103196477196219, 6.48522166387460142134390523002, 7.52310930683967015022869572164, 8.04802975984749778722505073447, 8.616240581857518025517515598181, 9.378736453100179638367289365414, 9.77705744325965679690367677767, 11.01679842517052806873527579504, 11.52875076356840618209482398120, 12.19979418065711379690085179709, 12.85420857700536218384931399117, 13.32531283515341655774591036731, 14.27053574840533226169863174046, 14.67787477080718393110157323210, 15.25348858658053440321890156248, 15.82470349198681761853983709853, 16.87516496585004159148553046451, 17.6023276651250336711211941490

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