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Properties

Label 1-112-112.13-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: $\chi_{112} (13, \cdot )$
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

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

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