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Properties

Label 2-111-111.29-c1-0-8
Degree $2$
Conductor $111$
Sign $0.745 + 0.666i$
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  + (0.448 + 0.120i)2-s + (1.01 − 1.40i)3-s + (−1.54 − 0.892i)4-s + (1.92 − 0.514i)5-s + (0.623 − 0.507i)6-s + (−1.43 + 2.48i)7-s + (−1.24 − 1.24i)8-s + (−0.941 − 2.84i)9-s + 0.923·10-s + 2.54·11-s + (−2.82 + 1.26i)12-s + (0.635 + 2.37i)13-s + (−0.943 + 0.943i)14-s + (1.22 − 3.22i)15-s + (1.37 + 2.38i)16-s + (−1.22 + 4.57i)17-s + ⋯
L(s)  = 1  + (0.316 + 0.0849i)2-s + (0.585 − 0.810i)3-s + (−0.772 − 0.446i)4-s + (0.859 − 0.230i)5-s + (0.254 − 0.207i)6-s + (−0.543 + 0.940i)7-s + (−0.439 − 0.439i)8-s + (−0.313 − 0.949i)9-s + 0.291·10-s + 0.766·11-s + (−0.814 + 0.364i)12-s + (0.176 + 0.658i)13-s + (−0.252 + 0.252i)14-s + (0.316 − 0.831i)15-s + (0.344 + 0.596i)16-s + (−0.297 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.745 + 0.666i$
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: \((2,\ 111,\ (\ :1/2),\ 0.745 + 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20462 - 0.459961i\)
\(L(\frac12)\) \(\approx\) \(1.20462 - 0.459961i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.01 + 1.40i)T \)
37 \( 1 + (-3.72 + 4.80i)T \)
good2 \( 1 + (-0.448 - 0.120i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.92 + 0.514i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.43 - 2.48i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.54T + 11T^{2} \)
13 \( 1 + (-0.635 - 2.37i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.22 - 4.57i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.827 - 3.08i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.53 + 4.53i)T + 23iT^{2} \)
29 \( 1 + (-2.84 + 2.84i)T - 29iT^{2} \)
31 \( 1 + (3.76 + 3.76i)T + 31iT^{2} \)
41 \( 1 + (-0.510 + 0.884i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.32 - 8.32i)T - 43iT^{2} \)
47 \( 1 + 2.59iT - 47T^{2} \)
53 \( 1 + (-12.2 + 7.06i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.02 + 3.80i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.538 - 0.144i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.48 - 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.4 + 6.00i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 + (0.754 + 2.81i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (14.5 - 8.37i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-10.3 - 2.77i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.01 - 3.01i)T - 97iT^{2} \)
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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

−13.49812397261124864128282398415, −12.79156185846129732610365343245, −11.90686855547051469456034106158, −9.929478471277676464827976712314, −9.181979001597650803041431235224, −8.370753524092136033384091091634, −6.36056173019305689173195899864, −5.88315265230560601910326285554, −3.92927714183948390652261294986, −1.92560940395088669389097645601, 3.03826242777685906290128356320, 4.15256437395138772523274218050, 5.46888040796426977497187224619, 7.19118992127769772255645002339, 8.655854059244158180629126169447, 9.598882553499570772962177959751, 10.26574596612481359499257998105, 11.70082984579285427255628690811, 13.29275523277008298573023958138, 13.70234272931038252375384518423

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