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Properties

Label 1-1200-1200.347-r1-0-0
Degree $1$
Conductor $1200$
Sign $-0.883 - 0.467i$
Analytic cond. $128.957$
Root an. cond. $128.957$
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  + i·7-s + (0.587 − 0.809i)11-s + (0.809 − 0.587i)13-s + (−0.951 − 0.309i)17-s + (−0.951 − 0.309i)19-s + (0.587 − 0.809i)23-s + (0.951 − 0.309i)29-s + (−0.309 + 0.951i)31-s + (0.809 − 0.587i)37-s + (−0.809 + 0.587i)41-s − 43-s + (−0.951 + 0.309i)47-s − 49-s + (−0.309 − 0.951i)53-s + (0.587 + 0.809i)59-s + ⋯
L(s)  = 1  + i·7-s + (0.587 − 0.809i)11-s + (0.809 − 0.587i)13-s + (−0.951 − 0.309i)17-s + (−0.951 − 0.309i)19-s + (0.587 − 0.809i)23-s + (0.951 − 0.309i)29-s + (−0.309 + 0.951i)31-s + (0.809 − 0.587i)37-s + (−0.809 + 0.587i)41-s − 43-s + (−0.951 + 0.309i)47-s − 49-s + (−0.309 − 0.951i)53-s + (0.587 + 0.809i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.883 - 0.467i$
Analytic conductor: \(128.957\)
Root analytic conductor: \(128.957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1200,\ (1:\ ),\ -0.883 - 0.467i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1295119992 - 0.5213821503i\)
\(L(\frac12)\) \(\approx\) \(0.1295119992 - 0.5213821503i\)
\(L(1)\) \(\approx\) \(0.9499082483 - 0.03989703524i\)
\(L(1)\) \(\approx\) \(0.9499082483 - 0.03989703524i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + iT \)
11 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (0.951 - 0.309i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.587 + 0.809i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
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

−21.32697606638264628316708525924, −20.37677058998140830293121605111, −19.938217081912395161551518168301, −19.08509851983979784458472926468, −18.231883159213215532620919224313, −17.22894140253629848712702145862, −16.97528202661576349048029996797, −15.91736826226690322684401948508, −15.099320648280133223290967055371, −14.35689982058278671318875599637, −13.4380785702072883437957377932, −12.96978989385899783507000921041, −11.81875560552230224799194528688, −11.11245199550805771666642083877, −10.339257371018933196967499573432, −9.48676305360027432320694582348, −8.64349818137869014419373642879, −7.73804452556258533614828629870, −6.70783977236268079001682493018, −6.37344420103603430682689474218, −4.875514631673893313675435336203, −4.187758099611233402394699044786, −3.45634768201283222684811210908, −2.00956748585041759177737237123, −1.25712707008780530928137297069, 0.10735382181168296187328908886, 1.30482853086599595141957514891, 2.49351549176946881274560630858, 3.24737134485929738934855840390, 4.38826813923981351683044388626, 5.29095302794428078897539543874, 6.29471095052565953842492783784, 6.71381969360040254626854109787, 8.32909009540129091357753358932, 8.56725877679719373380732219634, 9.420898709044843418697706333356, 10.60587017082632836681772167125, 11.22653331591465862851536040538, 12.021197882056053977821880870590, 12.945840399657308799398115470075, 13.5420282124353294845337093207, 14.63577638274025489275356413404, 15.20641869968614585969603587425, 16.067259853790215165373720954565, 16.68411190876329768536979152618, 17.87968789124917601884583283202, 18.19464994471158880367861578391, 19.26199138542301891699025478778, 19.70420798539906322942211608838, 20.811906995551942020880079025320

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