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Properties

Label 1-2925-2925.119-r0-0-0
Degree $1$
Conductor $2925$
Sign $-0.693 - 0.720i$
Analytic cond. $13.5836$
Root an. cond. $13.5836$
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  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.866 + 0.5i)7-s + (−0.951 + 0.309i)8-s + (0.587 + 0.809i)11-s + (0.104 + 0.994i)14-s + (−0.809 − 0.587i)16-s + (−0.669 + 0.743i)17-s + (−0.743 − 0.669i)19-s + (−0.309 + 0.951i)22-s + (−0.913 − 0.406i)23-s + (−0.743 + 0.669i)28-s + (−0.309 + 0.951i)29-s + (−0.207 + 0.978i)31-s − i·32-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.866 + 0.5i)7-s + (−0.951 + 0.309i)8-s + (0.587 + 0.809i)11-s + (0.104 + 0.994i)14-s + (−0.809 − 0.587i)16-s + (−0.669 + 0.743i)17-s + (−0.743 − 0.669i)19-s + (−0.309 + 0.951i)22-s + (−0.913 − 0.406i)23-s + (−0.743 + 0.669i)28-s + (−0.309 + 0.951i)29-s + (−0.207 + 0.978i)31-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.693 - 0.720i$
Analytic conductor: \(13.5836\)
Root analytic conductor: \(13.5836\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2925,\ (0:\ ),\ -0.693 - 0.720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4504959223 + 1.058159560i\)
\(L(\frac12)\) \(\approx\) \(-0.4504959223 + 1.058159560i\)
\(L(1)\) \(\approx\) \(0.8943333063 + 0.8145885168i\)
\(L(1)\) \(\approx\) \(0.8943333063 + 0.8145885168i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + (-0.913 - 0.406i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.207 + 0.978i)T \)
37 \( 1 + (-0.406 - 0.913i)T \)
41 \( 1 + (-0.994 - 0.104i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.207 - 0.978i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (-0.207 - 0.978i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (-0.994 + 0.104i)T \)
97 \( 1 + (-0.743 + 0.669i)T \)
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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

−18.74056265548839402593083731031, −18.23923295083857244302711277951, −17.26111361821631328346677462977, −16.75519854054290743399144606158, −15.60379092102918958570821369086, −15.05893424480042215134854035173, −14.123460313961741917705662627017, −13.82600361162111969720791136025, −13.15031496954696715143286963892, −12.11448653073332276718022691490, −11.54297965941537487925640630840, −11.07735096054661731537730260747, −10.25076351462372445426401422854, −9.567155113912562183966231092425, −8.653678234535228558115704621792, −7.999988736905593530451347308760, −6.90080276345524237731398050148, −6.07884079097549629766182820054, −5.40061474212419990745239537436, −4.38072586250303266804382658543, −4.01747571103612717171515746257, −3.061865993560371582744033733874, −2.04036801363908491663248857240, −1.412720470556341482801624850931, −0.25997256249197312671623186172, 1.67911769362933767845502268335, 2.31448065188127235528126675042, 3.51364447012160345493244373695, 4.300988494956390977165489168442, 4.90190169167536241633577002229, 5.6336479923914292628742627120, 6.62069220222020634815294962576, 7.00311176404698048237559340152, 8.07690856560624066347721754978, 8.616558841843536720638525609720, 9.20559401319849765429692391092, 10.3492147081312769628506384188, 11.21981788061869473440787703135, 11.97106724860690398111323385757, 12.58031125396418218738007549645, 13.214937063470300725201080017549, 14.24486225285090244648594154774, 14.6315960551957322272114572878, 15.25448907706999281266848672956, 15.87667251802217491446691550096, 16.790497552199333228053499274515, 17.42979074112922249282227080020, 17.93721677681108511188512727433, 18.5440135570932404019883384029, 19.805815793136098817645903426661

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