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Properties

Label 1-112-112.69-r1-0-0
Degree $1$
Conductor $112$
Sign $-0.923 + 0.382i$
Analytic cond. $12.0360$
Root an. cond. $12.0360$
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·3-s − i·5-s − 9-s + i·11-s + i·13-s + 15-s − 17-s + i·19-s − 23-s − 25-s − i·27-s − i·29-s − 31-s − 33-s + i·37-s + ⋯
L(s)  = 1  + i·3-s − i·5-s − 9-s + i·11-s + i·13-s + 15-s − 17-s + i·19-s − 23-s − 25-s − i·27-s − i·29-s − 31-s − 33-s + i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(12.0360\)
Root analytic conductor: \(12.0360\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 112,\ (1:\ ),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1522999074 + 0.7656633392i\)
\(L(\frac12)\) \(\approx\) \(0.1522999074 + 0.7656633392i\)
\(L(1)\) \(\approx\) \(0.7757132340 + 0.3213109420i\)
\(L(1)\) \(\approx\) \(0.7757132340 + 0.3213109420i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + T \)
5 \( 1 \)
11 \( 1 \)
13 \( 1 - iT \)
17 \( 1 \)
19 \( 1 \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 + T \)
59 \( 1 \)
61 \( 1 - T \)
67 \( 1 \)
71 \( 1 + iT \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 - T \)
97 \( 1 \)
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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

−29.04623484062898074119668505159, −27.69694966824933517836374117151, −26.47941981364820799085138990314, −25.68275864770198875830361366534, −24.52971713786062480127979137048, −23.71392848052880716591578568353, −22.54124450331906892672077537473, −21.811329931581130684507426919266, −20.06863679022408665972861293625, −19.327919819313878274225094654570, −18.1567897250983868455838807497, −17.676552208912830698880799709484, −16.0587216436009521726825830677, −14.76250922395474566375276906340, −13.77320253421672599446946618452, −12.842433040562791416456672273140, −11.41548016917890909094556269703, −10.67442415153499723216099602212, −8.87905611825583644799642840452, −7.67791594600339308771719279120, −6.648172831693047617475404408265, −5.60430011566691685658538347274, −3.40173645469644597001971836703, −2.25109987417179973074717490403, −0.30973555830596125055676388292, 1.970358489138820615405008870206, 4.02673280404834898486669986815, 4.738838081208124414868355336293, 6.1428756223057932463728723604, 7.97931265154033361691912723600, 9.18486072857960802604018187860, 9.91634880941144690182666898599, 11.39059252789262587739140370670, 12.41948705462780718047865575232, 13.80559945091482795518956623345, 14.99888386284970856349666740507, 16.06054838645802670088994559439, 16.81795784948575136152892763391, 17.89677921574138321151295201025, 19.60281924851378421847707092481, 20.45170448190696882440129318445, 21.22316940307112402178152994895, 22.31522223828657492455996048880, 23.39441074334576844402185230549, 24.532591290086086249919061067567, 25.6567284172290039631590032351, 26.601076161510604200291125378523, 27.72421636634409914293239416125, 28.397027364504276336492155349504, 29.24102808186841240518451722353

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