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Properties

Label 1-5600-5600.1747-r0-0-0
Degree $1$
Conductor $5600$
Sign $-0.250 + 0.968i$
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  + (−0.0523 − 0.998i)3-s + (−0.994 + 0.104i)9-s + (0.629 + 0.777i)11-s + (−0.156 + 0.987i)13-s + (0.207 + 0.978i)17-s + (−0.0523 + 0.998i)19-s + (−0.913 + 0.406i)23-s + (0.156 + 0.987i)27-s + (−0.453 + 0.891i)29-s + (0.978 − 0.207i)31-s + (0.743 − 0.669i)33-s + (0.629 − 0.777i)37-s + (0.994 + 0.104i)39-s + (−0.587 − 0.809i)41-s + (−0.707 − 0.707i)43-s + ⋯
L(s)  = 1  + (−0.0523 − 0.998i)3-s + (−0.994 + 0.104i)9-s + (0.629 + 0.777i)11-s + (−0.156 + 0.987i)13-s + (0.207 + 0.978i)17-s + (−0.0523 + 0.998i)19-s + (−0.913 + 0.406i)23-s + (0.156 + 0.987i)27-s + (−0.453 + 0.891i)29-s + (0.978 − 0.207i)31-s + (0.743 − 0.669i)33-s + (0.629 − 0.777i)37-s + (0.994 + 0.104i)39-s + (−0.587 − 0.809i)41-s + (−0.707 − 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $-0.250 + 0.968i$
Analytic conductor: \(26.0062\)
Root analytic conductor: \(26.0062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (1747, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (0:\ ),\ -0.250 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5440992300 + 0.7029529729i\)
\(L(\frac12)\) \(\approx\) \(0.5440992300 + 0.7029529729i\)
\(L(1)\) \(\approx\) \(0.9225563017 - 0.04366979995i\)
\(L(1)\) \(\approx\) \(0.9225563017 - 0.04366979995i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.0523 - 0.998i)T \)
11 \( 1 + (0.629 + 0.777i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
17 \( 1 + (0.207 + 0.978i)T \)
19 \( 1 + (-0.0523 + 0.998i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (-0.453 + 0.891i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.629 - 0.777i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (0.998 - 0.0523i)T \)
59 \( 1 + (0.933 - 0.358i)T \)
61 \( 1 + (-0.933 - 0.358i)T \)
67 \( 1 + (0.544 - 0.838i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.453 + 0.891i)T \)
89 \( 1 + (0.406 + 0.913i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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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

−17.57800065282157836480398004814, −16.882594157591104920553957948, −16.2778936189686236599144517135, −15.75747640462575608874122330689, −14.99694189227567089103465577778, −14.57852728910101127309365410559, −13.59145472009304106258615866754, −13.342017887695162410576039918593, −11.97930928709294118643168665422, −11.711534962432296488873957278218, −10.95847684542264221491467915056, −10.22799790375316678675831935267, −9.68204301963819434676563890390, −9.02117702273737500761737366337, −8.28725139286625901360762050358, −7.712108066769848077534884589875, −6.55364217327306655341837792638, −6.02393464665299266841342812281, −5.21651476714288913896078360752, −4.582097529734536854037286398674, −3.88594699577032462067011213083, −2.92257319613847665670102372103, −2.66940371051495301627473792336, −1.13894796171033518470197710883, −0.24270732241088408885955706870, 1.203245606341533815600049630581, 1.84617373568002484145661081851, 2.33046972443420701382513554337, 3.66996228527145888095165856715, 4.02952733429461261372401597415, 5.20963646270681097492966610375, 5.856167942755196915843820160945, 6.66556210113586346169363209200, 7.041236736075344438804225583813, 7.92937844904254489257503183044, 8.458010755865338523438850072558, 9.277397863316418114231137258101, 9.98408401194769710483763281466, 10.770752158400195403658728705884, 11.650233651572432115036939049326, 12.211480952657459359008606538740, 12.48476646818441268742317769740, 13.524756430493723829869790830530, 13.95895734260264899556700748017, 14.7180019084560000335546481059, 15.13407868721369231012434966642, 16.357916810565767200468756051163, 16.76940432276038375274855224160, 17.441404475105664048564358844161, 18.01165891506568404264139205094

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