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Properties

Label 1-5600-5600.4779-r0-0-0
Degree $1$
Conductor $5600$
Sign $-0.447 - 0.894i$
Analytic cond. $26.0062$
Root an. cond. $26.0062$
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  + (0.838 − 0.544i)3-s + (0.406 − 0.913i)9-s + (−0.933 − 0.358i)11-s + (0.156 − 0.987i)13-s + (−0.669 − 0.743i)17-s + (0.544 − 0.838i)19-s + (0.994 + 0.104i)23-s + (−0.156 − 0.987i)27-s + (0.891 + 0.453i)29-s + (0.669 + 0.743i)31-s + (−0.978 + 0.207i)33-s + (0.358 + 0.933i)37-s + (−0.406 − 0.913i)39-s + (−0.587 − 0.809i)41-s + (0.707 + 0.707i)43-s + ⋯
L(s)  = 1  + (0.838 − 0.544i)3-s + (0.406 − 0.913i)9-s + (−0.933 − 0.358i)11-s + (0.156 − 0.987i)13-s + (−0.669 − 0.743i)17-s + (0.544 − 0.838i)19-s + (0.994 + 0.104i)23-s + (−0.156 − 0.987i)27-s + (0.891 + 0.453i)29-s + (0.669 + 0.743i)31-s + (−0.978 + 0.207i)33-s + (0.358 + 0.933i)37-s + (−0.406 − 0.913i)39-s + (−0.587 − 0.809i)41-s + (0.707 + 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(26.0062\)
Root analytic conductor: \(26.0062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (4779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (0:\ ),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227754167 - 1.988379328i\)
\(L(\frac12)\) \(\approx\) \(1.227754167 - 1.988379328i\)
\(L(1)\) \(\approx\) \(1.279112145 - 0.5567951174i\)
\(L(1)\) \(\approx\) \(1.279112145 - 0.5567951174i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.838 - 0.544i)T \)
11 \( 1 + (-0.933 - 0.358i)T \)
13 \( 1 + (0.156 - 0.987i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.544 - 0.838i)T \)
23 \( 1 + (0.994 + 0.104i)T \)
29 \( 1 + (0.891 + 0.453i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.358 + 0.933i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (0.544 + 0.838i)T \)
59 \( 1 + (-0.629 + 0.777i)T \)
61 \( 1 + (-0.629 - 0.777i)T \)
67 \( 1 + (0.998 + 0.0523i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (-0.406 - 0.913i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.453 - 0.891i)T \)
89 \( 1 + (0.994 + 0.104i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−18.21190640951515684343105387059, −17.26891906159249727303647643923, −16.67820914436853964369066158718, −15.857719205971208833202538173446, −15.5051807227248860545314237029, −14.77933609939023235271654087377, −14.12954428664918510758343343797, −13.53158034617598065315228185912, −12.90878195086882573612305852843, −12.18571157117326573004278520998, −11.197300038345859139064459776745, −10.69104836481154557047398935168, −9.89375966694344428025093153036, −9.43584493301556252543176751160, −8.588437254486649355005568959635, −8.0929591430744050512002225940, −7.36465343740551835390617691590, −6.59862810552748863317098682500, −5.682580872519961298148387298916, −4.84224977323703185849821894353, −4.23609839411025033583936950258, −3.600455405758074659159196283389, −2.58791403164308462042898042460, −2.17806695234916567413376477220, −1.15085050975853678381389977899, 0.56550238519796355661993934263, 1.23684921155788142911090359871, 2.49457519945393981256492122079, 2.86050632081602585689338059922, 3.45837059887173892255037194822, 4.68705844574138166741479131316, 5.16822028320563656622137031480, 6.18448958191589522525433397484, 6.926036898570727900208908990787, 7.520620601549293430241641731637, 8.18630790431399754143454742111, 8.83868347677361663630974859242, 9.38038157359663083780063738242, 10.35824205872355786767186348714, 10.83254509712548515177625144480, 11.80098047483125464748082905229, 12.43352807557988875593518620714, 13.25968732062497518208227351212, 13.53129180155601831412780902976, 14.14960310772254267523020262582, 15.19043860224165753587978950246, 15.48891286701227232476676602308, 16.05889909936231773556798918592, 17.151859061534112580384913145064, 17.848701375862761605518575995664

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