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Properties

Label 1-123-123.14-r0-0-0
Degree $1$
Conductor $123$
Sign $0.652 + 0.758i$
Analytic cond. $0.571209$
Root an. cond. $0.571209$
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·2-s − 4-s + i·5-s + (−0.707 + 0.707i)7-s + i·8-s + 10-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + 16-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)19-s − i·20-s + (0.707 + 0.707i)22-s + 23-s + ⋯
L(s)  = 1  − i·2-s − 4-s + i·5-s + (−0.707 + 0.707i)7-s + i·8-s + 10-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + 16-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)19-s − i·20-s + (0.707 + 0.707i)22-s + 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(123\)    =    \(3 \cdot 41\)
Sign: $0.652 + 0.758i$
Analytic conductor: \(0.571209\)
Root analytic conductor: \(0.571209\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{123} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 123,\ (0:\ ),\ 0.652 + 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6087058185 + 0.2793572465i\)
\(L(\frac12)\) \(\approx\) \(0.6087058185 + 0.2793572465i\)
\(L(1)\) \(\approx\) \(0.7760683282 + 0.02292221188i\)
\(L(1)\) \(\approx\) \(0.7760683282 + 0.02292221188i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 \)
good2 \( 1 - iT \)
5 \( 1 + iT \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
43 \( 1 + iT \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 - T \)
61 \( 1 + iT \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (0.707 - 0.707i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−28.84174518914972025215732925137, −27.42101143965073941662740170041, −26.9243829369256144225065600405, −25.54451439223285038381862075878, −24.95864845769832763203716288166, −23.719831468652358069831364844746, −23.20775258892593438287195140990, −21.92174245676663693993718135402, −20.717684707553463051288803105653, −19.53582965144541803023655466904, −18.4313498403540453606758195369, −17.033565466204672164356719564903, −16.5395299515512062947417884648, −15.59287041752238480215399530108, −14.24629913266847823789576711179, −13.15420050003549869977539375202, −12.50173878112744710069064058282, −10.44396840467955093033869685460, −9.38368284142138100621719426347, −8.21726639562946660599469029058, −7.243809112421996570550219433731, −5.776230131310798648392039404706, −4.83907605902944026305887886046, −3.39033122192786780433033523795, −0.6255436853681027117329340585, 2.210539379953828696367844369788, 3.05624376258090891145387371063, 4.58428446718259362826437513311, 6.10928387818735542125364004436, 7.57200337359169835189440139085, 9.16239741994643621572285722660, 10.06375973077815599192632397482, 11.06552875773663158331799760676, 12.260643732245993327990756204119, 13.10711626204271015840684791150, 14.50123050784298378006632193740, 15.313529023251843686057898850018, 17.035553408861025792250304475, 18.18919583508943054334682841968, 19.01758364005561202212623328717, 19.6717247672796180509614811785, 21.24783109329996915011181279202, 21.81875414728820065511827012953, 22.85227240656047249616813517404, 23.62992395729605376509735136410, 25.483782841361552867364418756, 26.19473621521774658741427790039, 27.19959195341335663509482142085, 28.39890289155016189620070069120, 29.04249214741209536844910860388

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