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Properties

Label 2-183-183.50-c1-0-1
Degree $2$
Conductor $183$
Sign $0.334 - 0.942i$
Analytic cond. $1.46126$
Root an. cond. $1.20882$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.24 − 1.24i)2-s + (−0.576 + 1.63i)3-s + 1.10i·4-s + 1.59·5-s + (2.75 − 1.31i)6-s + (−2.44 + 2.44i)7-s + (−1.11 + 1.11i)8-s + (−2.33 − 1.88i)9-s + (−1.99 − 1.99i)10-s + (−2.99 + 2.99i)11-s + (−1.81 − 0.639i)12-s + 5.59·13-s + 6.09·14-s + (−0.920 + 2.60i)15-s + 4.98·16-s + (−3.60 + 3.60i)17-s + ⋯
L(s)  = 1  + (−0.881 − 0.881i)2-s + (−0.332 + 0.943i)3-s + 0.554i·4-s + 0.714·5-s + (1.12 − 0.538i)6-s + (−0.924 + 0.924i)7-s + (−0.392 + 0.392i)8-s + (−0.778 − 0.627i)9-s + (−0.629 − 0.629i)10-s + (−0.903 + 0.903i)11-s + (−0.523 − 0.184i)12-s + 1.55·13-s + 1.62·14-s + (−0.237 + 0.673i)15-s + 1.24·16-s + (−0.875 + 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $0.334 - 0.942i$
Analytic conductor: \(1.46126\)
Root analytic conductor: \(1.20882\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{183} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 183,\ (\ :1/2),\ 0.334 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431140 + 0.304429i\)
\(L(\frac12)\) \(\approx\) \(0.431140 + 0.304429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.576 - 1.63i)T \)
61 \( 1 + (-7.06 - 3.32i)T \)
good2 \( 1 + (1.24 + 1.24i)T + 2iT^{2} \)
5 \( 1 - 1.59T + 5T^{2} \)
7 \( 1 + (2.44 - 2.44i)T - 7iT^{2} \)
11 \( 1 + (2.99 - 2.99i)T - 11iT^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 + (3.60 - 3.60i)T - 17iT^{2} \)
19 \( 1 - 4.14iT - 19T^{2} \)
23 \( 1 + (-5.04 - 5.04i)T + 23iT^{2} \)
29 \( 1 + (-3.68 + 3.68i)T - 29iT^{2} \)
31 \( 1 + (6.69 + 6.69i)T + 31iT^{2} \)
37 \( 1 + (-0.644 - 0.644i)T + 37iT^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 + (0.0113 + 0.0113i)T + 43iT^{2} \)
47 \( 1 + 3.25iT - 47T^{2} \)
53 \( 1 + (-4.11 - 4.11i)T + 53iT^{2} \)
59 \( 1 + (-2.84 + 2.84i)T - 59iT^{2} \)
67 \( 1 + (2.80 + 2.80i)T + 67iT^{2} \)
71 \( 1 + (-8.53 + 8.53i)T - 71iT^{2} \)
73 \( 1 - 4.99T + 73T^{2} \)
79 \( 1 + (-1.30 + 1.30i)T - 79iT^{2} \)
83 \( 1 - 3.54iT - 83T^{2} \)
89 \( 1 + (5.67 - 5.67i)T - 89iT^{2} \)
97 \( 1 - 4.00iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.57814762898483201185087276954, −11.46534036108019241573857305152, −10.63226579958615383482861627487, −9.847004776810798611249588689290, −9.260919070596194874332101219518, −8.360667605170872718068338627632, −6.16429781213040353852345773169, −5.52305267304551688341768231595, −3.58712252495252144615491467627, −2.15213629796550760615087415170, 0.64650491901403359512313811000, 3.09047762063323046945872232014, 5.49346684363532569362570530630, 6.65983233742954651891094566543, 6.94578922267405008877373632401, 8.378821171544479753924315982741, 9.030092672284963540897079014743, 10.42909069175905061489031269784, 11.16590681981637530573643149625, 12.90662258481779995860316935334

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