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Properties

Label 1-60e2-3600.1411-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.563 + 0.826i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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  + (−0.5 − 0.866i)7-s + (−0.207 − 0.978i)11-s + (0.207 − 0.978i)13-s + (−0.809 + 0.587i)17-s + (0.587 + 0.809i)19-s + (0.669 + 0.743i)23-s + (−0.994 − 0.104i)29-s + (0.104 + 0.994i)31-s + (0.951 − 0.309i)37-s + (0.978 + 0.207i)41-s + (0.866 − 0.5i)43-s + (0.104 − 0.994i)47-s + (−0.5 + 0.866i)49-s + (−0.587 + 0.809i)53-s + (0.207 − 0.978i)59-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (−0.207 − 0.978i)11-s + (0.207 − 0.978i)13-s + (−0.809 + 0.587i)17-s + (0.587 + 0.809i)19-s + (0.669 + 0.743i)23-s + (−0.994 − 0.104i)29-s + (0.104 + 0.994i)31-s + (0.951 − 0.309i)37-s + (0.978 + 0.207i)41-s + (0.866 − 0.5i)43-s + (0.104 − 0.994i)47-s + (−0.5 + 0.866i)49-s + (−0.587 + 0.809i)53-s + (0.207 − 0.978i)59-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.563 + 0.826i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.563 + 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9638134677 + 0.5092365825i\)
\(L(\frac12)\) \(\approx\) \(0.9638134677 + 0.5092365825i\)
\(L(1)\) \(\approx\) \(0.9125802857 - 0.1217963492i\)
\(L(1)\) \(\approx\) \(0.9125802857 - 0.1217963492i\)

Euler product

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

−18.23488341092116293276543849653, −18.003852552272638266678107357720, −16.99559719815253153117737691228, −16.26452246359134484398093571252, −15.6919620223125717378135217580, −15.01068557548668894578929082346, −14.436874235887053969102546201580, −13.36408992923828082218927320573, −12.98487067690347591942089912022, −12.16696764998957086526658927986, −11.45522751319292631797341460089, −10.92882495430546268800831455973, −9.74641729723091793743182130216, −9.29258529295103655023902394669, −8.83711427918420978970282134954, −7.67705550160231147184318602974, −7.04687794551470822340922064737, −6.338827375216276496910501592582, −5.569617264973594124038104678662, −4.6175236677061744955745778120, −4.166468602805149490468301738528, −2.759097942640130642718282459822, −2.50030190471932618030388372506, −1.42901033698562091907521832529, −0.219999495650250675586717515779, 0.69520760086603389802528258520, 1.44334622856191258806256561018, 2.70287026011595179778118404323, 3.46478792962661142318438582540, 3.95722490379917186440788817688, 5.08202661000279716285100575040, 5.84500423439075809082014440361, 6.42386645242517039674886946465, 7.48792579882743495096144605887, 7.85727706659804660302487449131, 8.84865297742249253149475993074, 9.48914169881843080455759465063, 10.480192039315744172728723987, 10.786198722725057109883073201868, 11.54997758835541224799549104269, 12.62655546018282513223561500626, 13.12471442176059639864781153876, 13.69558118019946054513405921153, 14.43096444146690730968582799122, 15.273935427595193842978570568349, 15.99586956603338763349396978867, 16.47887120498215709815374045626, 17.32165369752022796987371701627, 17.80294568510427061056030365644, 18.75683620146807421447249853393

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