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Properties

Label 1-3744-3744.229-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.432 - 0.901i$
Analytic cond. $402.348$
Root an. cond. $402.348$
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  + (0.965 + 0.258i)5-s + (−0.5 + 0.866i)7-s + (−0.258 − 0.965i)11-s + 17-s + (0.707 + 0.707i)19-s + (0.866 − 0.5i)23-s + (0.866 + 0.5i)25-s + (−0.258 − 0.965i)29-s + (0.866 − 0.5i)31-s + (−0.707 + 0.707i)35-s + (0.707 − 0.707i)37-s + (−0.5 − 0.866i)41-s + (−0.965 + 0.258i)43-s + (−0.866 − 0.5i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)5-s + (−0.5 + 0.866i)7-s + (−0.258 − 0.965i)11-s + 17-s + (0.707 + 0.707i)19-s + (0.866 − 0.5i)23-s + (0.866 + 0.5i)25-s + (−0.258 − 0.965i)29-s + (0.866 − 0.5i)31-s + (−0.707 + 0.707i)35-s + (0.707 − 0.707i)37-s + (−0.5 − 0.866i)41-s + (−0.965 + 0.258i)43-s + (−0.866 − 0.5i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.432 - 0.901i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.432 - 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7787944039 - 1.236585671i\)
\(L(\frac12)\) \(\approx\) \(0.7787944039 - 1.236585671i\)
\(L(1)\) \(\approx\) \(1.152554992 + 0.03026104572i\)
\(L(1)\) \(\approx\) \(1.152554992 + 0.03026104572i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.258 - 0.965i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.965 + 0.258i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (-0.258 - 0.965i)T \)
67 \( 1 + (-0.258 + 0.965i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.965 + 0.258i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−18.51267940020534296193029422276, −17.92085414701856944206444806323, −17.244394313589565304995929133072, −16.72599190281984124509558426293, −16.093652785815401902218170232, −15.16806393633046246385061041123, −14.515275236237234743029994097956, −13.67454589582269663610365309400, −13.27226563714474946961439189610, −12.60868308009486601789169708337, −11.84916990527353257064103228304, −10.85678357805124905967545948160, −10.16507451926404009659938809761, −9.68128229107935531171492149774, −9.123047365864433032845901561205, −8.0469314702022449709593440648, −7.25868079434284016954825600856, −6.71937681812093559178598147483, −5.8818880463704889246826405715, −4.940811902739545637343796488174, −4.59640714004801308699084551141, −3.220609334347811795912857836743, −2.86646528721648675984493318080, −1.46374426939573851558400801943, −1.15923709908442457534230502404, 0.20865351720681117695554325904, 1.24990160487300514652726143940, 2.17908328637994417541987463732, 3.00967920092990037662865458880, 3.43751002784968332851118862444, 4.811454426757061623372372054300, 5.61818930999014333385375910275, 5.98082004535035770277096223218, 6.69307702724320143773942136395, 7.74215251344567806425756226721, 8.4527992447433628280431809224, 9.241700922624563694386904871223, 9.852575677973019480242568620703, 10.39977255807494003946687285543, 11.37475562612006574493275241613, 11.95844895675392706477702021496, 12.87645465935465083722680307181, 13.344742527844979162839815225279, 14.134806654959147461053917494092, 14.71309447748096860047824812859, 15.47701729436142186225553693884, 16.33234119306134622955793381957, 16.744204575569815241137159032094, 17.57763493518179863883601651607, 18.401630273996120478433482059376

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