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Properties

Label 2-111-111.29-c1-0-5
Degree $2$
Conductor $111$
Sign $-0.286 + 0.958i$
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  + (−2.10 − 0.563i)2-s + (0.615 − 1.61i)3-s + (2.37 + 1.36i)4-s + (1.20 − 0.321i)5-s + (−2.20 + 3.05i)6-s + (0.940 − 1.62i)7-s + (−1.13 − 1.13i)8-s + (−2.24 − 1.99i)9-s − 2.70·10-s − 2.84·11-s + (3.67 − 2.99i)12-s + (−0.693 − 2.58i)13-s + (−2.89 + 2.89i)14-s + (0.218 − 2.14i)15-s + (−0.990 − 1.71i)16-s + (1.15 − 4.32i)17-s + ⋯
L(s)  = 1  + (−1.48 − 0.398i)2-s + (0.355 − 0.934i)3-s + (1.18 + 0.684i)4-s + (0.536 − 0.143i)5-s + (−0.900 + 1.24i)6-s + (0.355 − 0.615i)7-s + (−0.401 − 0.401i)8-s + (−0.747 − 0.664i)9-s − 0.855·10-s − 0.857·11-s + (1.06 − 0.864i)12-s + (−0.192 − 0.717i)13-s + (−0.773 + 0.773i)14-s + (0.0563 − 0.552i)15-s + (−0.247 − 0.428i)16-s + (0.280 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.286 + 0.958i$
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: \((2,\ 111,\ (\ :1/2),\ -0.286 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.362930 - 0.487086i\)
\(L(\frac12)\) \(\approx\) \(0.362930 - 0.487086i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.615 + 1.61i)T \)
37 \( 1 + (-5.38 - 2.82i)T \)
good2 \( 1 + (2.10 + 0.563i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.20 + 0.321i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.940 + 1.62i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.84T + 11T^{2} \)
13 \( 1 + (0.693 + 2.58i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.15 + 4.32i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.28 - 4.80i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-6.37 - 6.37i)T + 23iT^{2} \)
29 \( 1 + (-2.64 + 2.64i)T - 29iT^{2} \)
31 \( 1 + (-0.558 - 0.558i)T + 31iT^{2} \)
41 \( 1 + (-3.42 + 5.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.64 + 4.64i)T - 43iT^{2} \)
47 \( 1 - 2.51iT - 47T^{2} \)
53 \( 1 + (1.75 - 1.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.01 - 11.2i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (10.5 - 2.83i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-11.9 - 6.90i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.21 + 5.32i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.59iT - 73T^{2} \)
79 \( 1 + (-2.61 - 9.74i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (3.19 - 1.84i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.11 - 1.37i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.49 + 4.49i)T - 97iT^{2} \)
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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

−13.35461606490052445983850798909, −12.10635866569835230877329259420, −11.03264019174626067827530680106, −9.980960852095192607901384264169, −9.082474202792987420580334284085, −7.75835659120890667989460275033, −7.44488958962178196177290851438, −5.54408093439056965769101694403, −2.76492820278124907664920330849, −1.16445055836115982615205986673, 2.44522082378772490543267560916, 4.79052875639545569887776132899, 6.30845892520917099806275809882, 7.84222492887541554277908427477, 8.768751562305401244402721461786, 9.500235259673415634709076022365, 10.47441219329629335708395691997, 11.21542535445398370955895416178, 12.99541929132001234697470296202, 14.38064160436759732897911150465

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