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Properties

Label 2-111-111.23-c1-0-6
Degree $2$
Conductor $111$
Sign $0.999 - 0.0192i$
Analytic cond. $0.886339$
Root an. cond. $0.941456$
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  + (2.05 − 0.550i)2-s + (−1.12 + 1.31i)3-s + (2.18 − 1.26i)4-s + (1.01 + 0.271i)5-s + (−1.59 + 3.32i)6-s + (−0.456 − 0.791i)7-s + (0.790 − 0.790i)8-s + (−0.456 − 2.96i)9-s + 2.23·10-s − 4.31·11-s + (−0.806 + 4.29i)12-s + (1.38 − 5.16i)13-s + (−1.37 − 1.37i)14-s + (−1.50 + 1.02i)15-s + (−1.33 + 2.31i)16-s + (1.19 + 4.45i)17-s + ⋯
L(s)  = 1  + (1.45 − 0.389i)2-s + (−0.651 + 0.758i)3-s + (1.09 − 0.631i)4-s + (0.453 + 0.121i)5-s + (−0.650 + 1.35i)6-s + (−0.172 − 0.299i)7-s + (0.279 − 0.279i)8-s + (−0.152 − 0.988i)9-s + 0.706·10-s − 1.30·11-s + (−0.232 + 1.24i)12-s + (0.383 − 1.43i)13-s + (−0.367 − 0.367i)14-s + (−0.387 + 0.265i)15-s + (−0.334 + 0.578i)16-s + (0.289 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.999 - 0.0192i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :1/2),\ 0.999 - 0.0192i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70430 + 0.0163644i\)
\(L(\frac12)\) \(\approx\) \(1.70430 + 0.0163644i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.12 - 1.31i)T \)
37 \( 1 + (-5.83 - 1.72i)T \)
good2 \( 1 + (-2.05 + 0.550i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.01 - 0.271i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.456 + 0.791i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.31T + 11T^{2} \)
13 \( 1 + (-1.38 + 5.16i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.19 - 4.45i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.494 - 1.84i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.392 - 0.392i)T - 23iT^{2} \)
29 \( 1 + (-0.841 - 0.841i)T + 29iT^{2} \)
31 \( 1 + (-4.29 + 4.29i)T - 31iT^{2} \)
41 \( 1 + (-3.89 - 6.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.76 - 1.76i)T + 43iT^{2} \)
47 \( 1 + 6.56iT - 47T^{2} \)
53 \( 1 + (8.17 + 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.21 - 12.0i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.55 - 0.953i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (13.5 - 7.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.61 + 1.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.61iT - 73T^{2} \)
79 \( 1 + (0.516 - 1.92i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (13.7 + 7.95i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-16.1 + 4.32i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.01 + 5.01i)T + 97iT^{2} \)
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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

−13.33369342618127568916233218118, −12.86407919308482268698272890784, −11.70271731085797155184198892072, −10.55870650901369388590395809855, −10.10760050508989849298086447848, −8.138811999371731035934090171896, −6.08961347046433725692546963207, −5.54568673954015259315950037581, −4.24310813914733787800985916666, −2.96974477746067447444876196084, 2.55388603200264002221405424989, 4.65069654166338356844315564810, 5.61768811988529396231148216529, 6.53642665929059010798629965627, 7.61471223533516285954662694203, 9.381465560450325442791936295425, 11.01726317867404949113660467008, 11.98427331021354366508560690571, 12.80375320866689658465508913058, 13.67143004897528797159717536895

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