Properties

Label 2-111-111.59-c1-0-1
Degree $2$
Conductor $111$
Sign $0.287 - 0.957i$
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  + (0.420 + 0.902i)2-s + (−1.71 + 0.272i)3-s + (0.648 − 0.772i)4-s + (1.24 + 1.77i)5-s + (−0.965 − 1.42i)6-s + (−0.867 + 4.92i)7-s + (2.89 + 0.775i)8-s + (2.85 − 0.933i)9-s + (−1.08 + 1.87i)10-s + (−2.36 − 4.09i)11-s + (−0.898 + 1.49i)12-s + (−0.286 − 3.27i)13-s + (−4.80 + 1.28i)14-s + (−2.61 − 2.70i)15-s + (0.167 + 0.950i)16-s + (−0.0360 + 0.412i)17-s + ⋯
L(s)  = 1  + (0.297 + 0.638i)2-s + (−0.987 + 0.157i)3-s + (0.324 − 0.386i)4-s + (0.557 + 0.795i)5-s + (−0.394 − 0.583i)6-s + (−0.328 + 1.86i)7-s + (1.02 + 0.274i)8-s + (0.950 − 0.311i)9-s + (−0.341 + 0.592i)10-s + (−0.712 − 1.23i)11-s + (−0.259 + 0.432i)12-s + (−0.0794 − 0.907i)13-s + (−1.28 + 0.344i)14-s + (−0.675 − 0.697i)15-s + (0.0419 + 0.237i)16-s + (−0.00874 + 0.0999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.287 - 0.957i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :1/2),\ 0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851087 + 0.632962i\)
\(L(\frac12)\) \(\approx\) \(0.851087 + 0.632962i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.272i)T \)
37 \( 1 + (3.19 - 5.17i)T \)
good2 \( 1 + (-0.420 - 0.902i)T + (-1.28 + 1.53i)T^{2} \)
5 \( 1 + (-1.24 - 1.77i)T + (-1.71 + 4.69i)T^{2} \)
7 \( 1 + (0.867 - 4.92i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (2.36 + 4.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.286 + 3.27i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (0.0360 - 0.412i)T + (-16.7 - 2.95i)T^{2} \)
19 \( 1 + (-1.79 - 0.837i)T + (12.2 + 14.5i)T^{2} \)
23 \( 1 + (1.00 + 3.76i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.994 + 3.71i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.758 - 0.758i)T - 31iT^{2} \)
41 \( 1 + (0.196 + 0.165i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (4.12 + 4.12i)T + 43iT^{2} \)
47 \( 1 + (-7.62 - 4.40i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.36 - 0.241i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.237 + 0.166i)T + (20.1 + 55.4i)T^{2} \)
61 \( 1 + (-9.17 + 0.802i)T + (60.0 - 10.5i)T^{2} \)
67 \( 1 + (-0.0440 - 0.00775i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-2.72 - 7.47i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 - 0.498iT - 73T^{2} \)
79 \( 1 + (5.22 - 3.65i)T + (27.0 - 74.2i)T^{2} \)
83 \( 1 + (5.58 + 6.65i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (8.80 - 12.5i)T + (-30.4 - 83.6i)T^{2} \)
97 \( 1 + (8.93 - 2.39i)T + (84.0 - 48.5i)T^{2} \)
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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

−14.03404468159092668510819331846, −12.82045947589014240398948873448, −11.70442432260424171657858742617, −10.70918579627786423964600489956, −9.935651373010826401563770965160, −8.285287117017266259879558521160, −6.63005598765078559949521458423, −5.81693346800679813413426257669, −5.36045643867572416001636919182, −2.66395167889334583167613244754, 1.56718582583101426193291069981, 4.04594445907512982668054288155, 5.01308048409491875019622224920, 6.93285651131505091355757414768, 7.46929348271532678580201985396, 9.703732187434373964256872353674, 10.44029804669099337887422525635, 11.40480549800178732643434281545, 12.51810131852131678463992740769, 13.13176643057908948197968648659

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