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Properties

Label 1-5600-5600.1419-r0-0-0
Degree $1$
Conductor $5600$
Sign $-0.823 - 0.567i$
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  + (−0.998 − 0.0523i)3-s + (0.994 + 0.104i)9-s + (−0.629 + 0.777i)11-s + (0.987 − 0.156i)13-s + (0.978 + 0.207i)17-s + (0.0523 + 0.998i)19-s + (0.406 − 0.913i)23-s + (−0.987 − 0.156i)27-s + (−0.453 − 0.891i)29-s + (−0.978 − 0.207i)31-s + (0.669 − 0.743i)33-s + (−0.777 + 0.629i)37-s + (−0.994 + 0.104i)39-s + (0.587 − 0.809i)41-s + (0.707 + 0.707i)43-s + ⋯
L(s)  = 1  + (−0.998 − 0.0523i)3-s + (0.994 + 0.104i)9-s + (−0.629 + 0.777i)11-s + (0.987 − 0.156i)13-s + (0.978 + 0.207i)17-s + (0.0523 + 0.998i)19-s + (0.406 − 0.913i)23-s + (−0.987 − 0.156i)27-s + (−0.453 − 0.891i)29-s + (−0.978 − 0.207i)31-s + (0.669 − 0.743i)33-s + (−0.777 + 0.629i)37-s + (−0.994 + 0.104i)39-s + (0.587 − 0.809i)41-s + (0.707 + 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $-0.823 - 0.567i$
Analytic conductor: \(26.0062\)
Root analytic conductor: \(26.0062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (1419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (0:\ ),\ -0.823 - 0.567i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06989463160 - 0.2244904383i\)
\(L(\frac12)\) \(\approx\) \(0.06989463160 - 0.2244904383i\)
\(L(1)\) \(\approx\) \(0.7039748546 + 0.005396714138i\)
\(L(1)\) \(\approx\) \(0.7039748546 + 0.005396714138i\)

Euler product

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

−18.03756891002370736937588503413, −17.54325533960488673042314477412, −16.61640293243630147757502520762, −16.25722361112725516543791008970, −15.6869311190824132968276862338, −14.97661420904725294101296927352, −14.02023274601279038816852938410, −13.3616822117253477462833132027, −12.84572973316193382496787764833, −12.04915875477166158342864278849, −11.355925788794008948998632605779, −10.81598027558542063104249232131, −10.41940449426851377020733284411, −9.27249380069526116854891185671, −8.94875307209496131866529859555, −7.7524099664754998765234556210, −7.316342251877062427346214509455, −6.44229478634308330257290860388, −5.67225894208331983895463401130, −5.34493914346285483017310675989, −4.47632809530828045858947117941, −3.53716355705113096552876672753, −2.99370364932887334755507511252, −1.650345460725620049749002913591, −1.03829249381625566524451004093, 0.07862435054336287679178782642, 1.274423640114434822626328504504, 1.84225385093231682465817962896, 3.001015083628374568933609462122, 3.88980031285080591200382033375, 4.54287746173204997366451588026, 5.40265888794938787219701030068, 5.89945584845996371117903615112, 6.55212959427506763612003540416, 7.508077685259943704940068440835, 7.873357064975071940500594663, 8.86999563271409470506048850195, 9.78254756553710922219256360383, 10.34440998687552308885972399366, 10.8311817940254882550770346185, 11.63208976089754992109958201864, 12.29350556442699209788575088493, 12.84208215145751618915204704648, 13.36802831114450967762365137658, 14.40085142095748678464791513345, 14.98510613142931046394729972320, 15.80282126061825835368332133023, 16.29844621738383166997595147144, 16.890589370552304614705837294502, 17.600104829837090128791969505985

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