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Properties

Label 1-60e2-3600.173-r0-0-0
Degree $1$
Conductor $3600$
Sign $-0.563 - 0.826i$
Analytic cond. $16.7183$
Root an. cond. $16.7183$
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  + (0.866 − 0.5i)7-s + (−0.207 − 0.978i)11-s + (0.978 + 0.207i)13-s + (−0.587 − 0.809i)17-s + (0.587 + 0.809i)19-s + (−0.743 + 0.669i)23-s + (−0.994 − 0.104i)29-s + (−0.104 − 0.994i)31-s + (−0.309 − 0.951i)37-s + (−0.978 − 0.207i)41-s + (−0.5 − 0.866i)43-s + (−0.994 − 0.104i)47-s + (0.5 − 0.866i)49-s + (0.809 + 0.587i)53-s + (−0.207 + 0.978i)59-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (−0.207 − 0.978i)11-s + (0.978 + 0.207i)13-s + (−0.587 − 0.809i)17-s + (0.587 + 0.809i)19-s + (−0.743 + 0.669i)23-s + (−0.994 − 0.104i)29-s + (−0.104 − 0.994i)31-s + (−0.309 − 0.951i)37-s + (−0.978 − 0.207i)41-s + (−0.5 − 0.866i)43-s + (−0.994 − 0.104i)47-s + (0.5 − 0.866i)49-s + (0.809 + 0.587i)53-s + (−0.207 + 0.978i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.563 - 0.826i$
Analytic conductor: \(16.7183\)
Root analytic conductor: \(16.7183\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (0:\ ),\ -0.563 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5978129972 - 1.131458810i\)
\(L(\frac12)\) \(\approx\) \(0.5978129972 - 1.131458810i\)
\(L(1)\) \(\approx\) \(1.027842308 - 0.2586566301i\)
\(L(1)\) \(\approx\) \(1.027842308 - 0.2586566301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.207 - 0.978i)T \)
13 \( 1 + (0.978 + 0.207i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (-0.743 + 0.669i)T \)
29 \( 1 + (-0.994 - 0.104i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.207 + 0.978i)T \)
61 \( 1 + (-0.207 - 0.978i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.994 + 0.104i)T \)
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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

−18.755156583606396489118189950730, −18.02485517401033394773965120515, −17.84462523349062743223054343005, −16.97392626427641856133805709452, −16.04838099339449181715274197266, −15.44485279227846742625194160270, −14.866144043910552136722456861487, −14.24371979125393160168803516709, −13.2312243805115242893006497554, −12.871474220360925901977160020618, −11.810235465065725177665655129459, −11.43407918407753336721056416462, −10.53418323524579697356559084543, −9.964505347251074810062297631331, −8.88756153630981518566262785550, −8.47505003642874215789715962453, −7.711501839006695910557922652516, −6.84590200602389737878695988956, −6.1320340550245328574722192225, −5.19657495617593978282897908915, −4.67058611739544473188333096770, −3.79587615767617962993099433505, −2.83593134106151107625668509879, −1.89222987822960960707268266141, −1.34340327079647313743883858114, 0.352705220411863876646878734725, 1.45496667725207641578485995189, 2.11852717336729500845192551016, 3.428598571390670496036633521622, 3.81918788838388835637679283439, 4.8291681169871262459117197133, 5.62117753217227825716575234815, 6.19948750617435970676211443609, 7.31699392294584289075856408505, 7.79255794197423999764456737399, 8.599709467750980971879504166335, 9.21711205877585774403636822603, 10.19400819159390059266461106511, 10.87869229968238886448259402258, 11.50526552253386669889015504308, 11.94115386272985282262831104695, 13.23518749477323781060641701971, 13.66311711579447672212161383423, 14.12178561291323308720852046098, 15.0523656612658902612688325790, 15.75310118272613385052830484564, 16.492333779899169172250185786676, 16.9286833357035287053751555056, 18.0883765970379640369026014036, 18.21373578799617154162740295329

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