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Properties

Label 2-111-111.17-c1-0-1
Degree $2$
Conductor $111$
Sign $0.686 - 0.727i$
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  + (−1.70 − 0.300i)3-s + (0.684 + 1.87i)4-s + (3.54 + 2.97i)7-s + (2.81 + 1.02i)9-s + (−0.601 − 3.41i)12-s + (−4.92 − 2.29i)13-s + (−3.06 + 2.57i)16-s + (5.59 − 3.91i)19-s + (−5.14 − 6.13i)21-s + (−4.92 + 0.868i)25-s + (−4.49 − 2.59i)27-s + (−3.16 + 8.68i)28-s + (7.07 − 7.07i)31-s + 6i·36-s + (0.5 − 6.06i)37-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (0.342 + 0.939i)4-s + (1.33 + 1.12i)7-s + (0.939 + 0.342i)9-s + (−0.173 − 0.984i)12-s + (−1.36 − 0.636i)13-s + (−0.766 + 0.642i)16-s + (1.28 − 0.898i)19-s + (−1.12 − 1.33i)21-s + (−0.984 + 0.173i)25-s + (−0.866 − 0.499i)27-s + (−0.597 + 1.64i)28-s + (1.27 − 1.27i)31-s + i·36-s + (0.0821 − 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.686 - 0.727i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :1/2),\ 0.686 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.834667 + 0.359849i\)
\(L(\frac12)\) \(\approx\) \(0.834667 + 0.359849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.300i)T \)
37 \( 1 + (-0.5 + 6.06i)T \)
good2 \( 1 + (-0.684 - 1.87i)T^{2} \)
5 \( 1 + (4.92 - 0.868i)T^{2} \)
7 \( 1 + (-3.54 - 2.97i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.92 + 2.29i)T + (8.35 + 9.95i)T^{2} \)
17 \( 1 + (10.9 - 13.0i)T^{2} \)
19 \( 1 + (-5.59 + 3.91i)T + (6.49 - 17.8i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-7.07 + 7.07i)T - 31iT^{2} \)
41 \( 1 + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.418 + 0.418i)T + 43iT^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-58.1 - 10.2i)T^{2} \)
61 \( 1 + (2.11 - 4.53i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (0.406 - 0.484i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (66.7 + 24.2i)T^{2} \)
73 \( 1 + 15.3iT - 73T^{2} \)
79 \( 1 + (-1.03 - 11.8i)T + (-77.7 + 13.7i)T^{2} \)
83 \( 1 + (-63.5 - 53.3i)T^{2} \)
89 \( 1 + (87.6 + 15.4i)T^{2} \)
97 \( 1 + (18.7 - 5.02i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.55238179706998140010041083206, −12.30988288077902524290255645655, −11.82658971363686989810479584865, −11.10073535162251497466846333354, −9.549770740789185436210440299517, −8.035748708079066621519372931513, −7.32122619045076470109185671744, −5.68701444751581782075563380553, −4.66995234894884843109210553614, −2.39791196442862499877901612725, 1.41362410418736017032243944118, 4.49382623556398749367053722803, 5.30468247137975634135482509705, 6.76001763904772478691091214890, 7.71278572501007803617164663854, 9.796592890473529980192113423759, 10.35116345101387706689873210892, 11.46250471174666724525215288231, 11.99669977500919616347521794766, 13.84361114920024322203055113484

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