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Properties

Label 1-5600-5600.859-r0-0-0
Degree $1$
Conductor $5600$
Sign $-0.503 - 0.864i$
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  + (−0.777 − 0.629i)3-s + (0.207 + 0.978i)9-s + (−0.544 − 0.838i)11-s + (−0.453 − 0.891i)13-s + (−0.913 + 0.406i)17-s + (0.629 + 0.777i)19-s + (−0.743 − 0.669i)23-s + (0.453 − 0.891i)27-s + (0.156 − 0.987i)29-s + (0.913 − 0.406i)31-s + (−0.104 + 0.994i)33-s + (0.838 + 0.544i)37-s + (−0.207 + 0.978i)39-s + (0.951 + 0.309i)41-s + (−0.707 − 0.707i)43-s + ⋯
L(s)  = 1  + (−0.777 − 0.629i)3-s + (0.207 + 0.978i)9-s + (−0.544 − 0.838i)11-s + (−0.453 − 0.891i)13-s + (−0.913 + 0.406i)17-s + (0.629 + 0.777i)19-s + (−0.743 − 0.669i)23-s + (0.453 − 0.891i)27-s + (0.156 − 0.987i)29-s + (0.913 − 0.406i)31-s + (−0.104 + 0.994i)33-s + (0.838 + 0.544i)37-s + (−0.207 + 0.978i)39-s + (0.951 + 0.309i)41-s + (−0.707 − 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $-0.503 - 0.864i$
Analytic conductor: \(26.0062\)
Root analytic conductor: \(26.0062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (0:\ ),\ -0.503 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4589519478 - 0.7983395904i\)
\(L(\frac12)\) \(\approx\) \(0.4589519478 - 0.7983395904i\)
\(L(1)\) \(\approx\) \(0.7210987849 - 0.2463992983i\)
\(L(1)\) \(\approx\) \(0.7210987849 - 0.2463992983i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.777 - 0.629i)T \)
11 \( 1 + (-0.544 - 0.838i)T \)
13 \( 1 + (-0.453 - 0.891i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.629 + 0.777i)T \)
23 \( 1 + (-0.743 - 0.669i)T \)
29 \( 1 + (0.156 - 0.987i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.838 + 0.544i)T \)
41 \( 1 + (0.951 + 0.309i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (0.629 - 0.777i)T \)
59 \( 1 + (-0.0523 + 0.998i)T \)
61 \( 1 + (-0.0523 - 0.998i)T \)
67 \( 1 + (-0.358 + 0.933i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (-0.207 + 0.978i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (0.987 - 0.156i)T \)
89 \( 1 + (-0.743 - 0.669i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−17.936614937444596897694691752964, −17.53460442854445858877131659859, −16.62154820129593790507641843859, −16.139900009594510249727210063990, −15.468758269605888494115136461624, −15.03412439853638127276440431950, −14.09150426105280966623028846106, −13.48907256195937875695107832419, −12.55836971685750778818397871748, −12.02564752740441573085341711200, −11.40779946943465510214064941290, −10.755322448005258962116267111066, −10.08042630382392131833888304414, −9.3263013169833234105968467442, −9.04674320808204149837359985325, −7.78318001174748778950454058810, −7.113721986425657806243198410574, −6.53355478931257585245301283847, −5.70421939915987610984502618157, −4.76352853227674394390844131457, −4.65277795580520488088199876846, −3.68766715263235231868713842984, −2.71347694831899009628131828676, −1.93262843806336700584087718595, −0.81016513730467756945016389111, 0.373623640792158763400737860217, 1.0881405353635817466349889302, 2.27051912937306387203600151221, 2.71809115900702960991530993131, 3.89765434874197572825972659436, 4.65155043767293287948412062590, 5.52098646121627572610254802405, 5.986311137740179193065837233627, 6.596533055140268536184223731154, 7.5891320823634976567861440841, 8.04075737920462681836361396354, 8.62538712464519018499149603178, 9.93459216839672485429967105207, 10.24455864668903127977765701035, 11.09156910775250795505295225188, 11.62459951787005845357018353027, 12.34262751750323400555633289460, 12.956026105306458526519227779854, 13.55326800563037651980633634859, 14.119465789779304969148852690198, 15.13243244852053477237134423531, 15.74155030665327999569397712485, 16.401677072733160314900817304747, 17.02574484933921831963575140541, 17.66668493100532043204179210292

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