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Properties

Label 2-111-111.29-c1-0-2
Degree $2$
Conductor $111$
Sign $0.996 - 0.0810i$
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  + (−1.28 − 0.345i)2-s + (1.34 + 1.09i)3-s + (−0.190 − 0.109i)4-s + (1.53 − 0.410i)5-s + (−1.34 − 1.87i)6-s + (0.317 − 0.549i)7-s + (2.09 + 2.09i)8-s + (0.595 + 2.94i)9-s − 2.11·10-s + 2.64·11-s + (−0.134 − 0.355i)12-s + (−0.262 − 0.980i)13-s + (−0.598 + 0.598i)14-s + (2.50 + 1.12i)15-s + (−1.75 − 3.04i)16-s + (−0.0282 + 0.105i)17-s + ⋯
L(s)  = 1  + (−0.911 − 0.244i)2-s + (0.774 + 0.633i)3-s + (−0.0950 − 0.0548i)4-s + (0.684 − 0.183i)5-s + (−0.550 − 0.766i)6-s + (0.119 − 0.207i)7-s + (0.740 + 0.740i)8-s + (0.198 + 0.980i)9-s − 0.668·10-s + 0.796·11-s + (−0.0388 − 0.102i)12-s + (−0.0728 − 0.271i)13-s + (−0.159 + 0.159i)14-s + (0.646 + 0.291i)15-s + (−0.439 − 0.760i)16-s + (−0.00684 + 0.0255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.996 - 0.0810i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :1/2),\ 0.996 - 0.0810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.873908 + 0.0354853i\)
\(L(\frac12)\) \(\approx\) \(0.873908 + 0.0354853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 - 1.09i)T \)
37 \( 1 + (5.94 - 1.27i)T \)
good2 \( 1 + (1.28 + 0.345i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.53 + 0.410i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.317 + 0.549i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.64T + 11T^{2} \)
13 \( 1 + (0.262 + 0.980i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.0282 - 0.105i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.101 + 0.378i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.15 + 5.15i)T + 23iT^{2} \)
29 \( 1 + (6.05 - 6.05i)T - 29iT^{2} \)
31 \( 1 + (2.83 + 2.83i)T + 31iT^{2} \)
41 \( 1 + (-5.64 + 9.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.03 + 5.03i)T - 43iT^{2} \)
47 \( 1 - 1.67iT - 47T^{2} \)
53 \( 1 + (7.72 - 4.45i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.786 + 2.93i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.90 + 2.11i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.13 + 0.654i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.65 - 4.99i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.10iT - 73T^{2} \)
79 \( 1 + (3.97 + 14.8i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (9.84 - 5.68i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.31 + 1.15i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.84 - 5.84i)T - 97iT^{2} \)
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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

−14.05754135311625578072774167120, −12.74476889091692789139322817940, −11.05329019008217391172991701448, −10.21389709222737830069880362233, −9.338085224444572564967496838494, −8.693718818937065069499745038297, −7.47881575218257592325362096321, −5.51115835590138434970464221941, −4.07020348126642229778379620536, −1.97096839719397357650398223287, 1.80669008392590398907406329891, 3.87552632817278643976344417262, 6.10996072656143180155289381910, 7.30467666054438976042573524979, 8.258391478038863417480770900526, 9.344823668709834594397612168500, 9.842653922599022825051454065862, 11.56992545288528220998749622871, 12.80486835529450399925228000466, 13.73086128596653266288950676873

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