Properties

Label 2-111-111.14-c1-0-9
Degree $2$
Conductor $111$
Sign $-0.997 - 0.0675i$
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  + (−0.683 − 2.55i)2-s + (0.839 − 1.51i)3-s + (−4.30 + 2.48i)4-s + (−0.283 + 1.05i)5-s + (−4.43 − 1.10i)6-s + (−1.64 − 2.85i)7-s + (5.55 + 5.55i)8-s + (−1.58 − 2.54i)9-s + 2.89·10-s + 1.47·11-s + (0.150 + 8.61i)12-s + (3.81 + 1.02i)13-s + (−6.16 + 6.16i)14-s + (1.36 + 1.31i)15-s + (5.39 − 9.33i)16-s + (6.35 − 1.70i)17-s + ⋯
L(s)  = 1  + (−0.483 − 1.80i)2-s + (0.484 − 0.874i)3-s + (−2.15 + 1.24i)4-s + (−0.126 + 0.473i)5-s + (−1.81 − 0.451i)6-s + (−0.623 − 1.07i)7-s + (1.96 + 1.96i)8-s + (−0.529 − 0.848i)9-s + 0.915·10-s + 0.444·11-s + (0.0434 + 2.48i)12-s + (1.05 + 0.283i)13-s + (−1.64 + 1.64i)14-s + (0.352 + 0.340i)15-s + (1.34 − 2.33i)16-s + (1.54 − 0.413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.997 - 0.0675i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :1/2),\ -0.997 - 0.0675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0268158 + 0.792910i\)
\(L(\frac12)\) \(\approx\) \(0.0268158 + 0.792910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.839 + 1.51i)T \)
37 \( 1 + (2.43 - 5.57i)T \)
good2 \( 1 + (0.683 + 2.55i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.283 - 1.05i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.64 + 2.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + (-3.81 - 1.02i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-6.35 + 1.70i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.74 + 1.27i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.17 + 2.17i)T + 23iT^{2} \)
29 \( 1 + (0.685 - 0.685i)T - 29iT^{2} \)
31 \( 1 + (-1.98 - 1.98i)T + 31iT^{2} \)
41 \( 1 + (-2.55 - 4.42i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.78 + 2.78i)T - 43iT^{2} \)
47 \( 1 - 6.97iT - 47T^{2} \)
53 \( 1 + (1.68 + 0.975i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.35 + 1.43i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.141 - 0.526i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.22 - 3.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.74 - 1.58i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.709iT - 73T^{2} \)
79 \( 1 + (-7.13 - 1.91i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-12.7 - 7.37i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.26 + 12.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.53 + 2.53i)T - 97iT^{2} \)
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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

−12.93078976261645617707606686842, −12.04670386961178793090251829734, −11.02083819076341682405984868619, −10.13688361709161379487088450115, −9.048269098112056155182228078568, −7.972102288130768192834327159543, −6.65364859300555166858401928683, −3.93017628078033332703435243264, −3.00146600663723215958012812735, −1.16174943886522158047001995262, 3.90115498925025437761126477261, 5.43684271475029158307385824838, 6.17643097476195863484919833269, 7.938338151255012567652326069741, 8.690373795457559610514157572764, 9.368077325480789208974749239747, 10.43818887243817782553189455752, 12.41653980138760603828264726853, 13.65302168223783440810239039163, 14.68984504226460510533024943902

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