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Properties

Label 2-111-37.7-c1-0-3
Degree $2$
Conductor $111$
Sign $0.871 + 0.490i$
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  + (−2.06 + 0.753i)2-s + (0.939 + 0.342i)3-s + (2.18 − 1.83i)4-s + (−0.647 − 3.67i)5-s − 2.20·6-s + (−0.172 − 0.977i)7-s + (−0.936 + 1.62i)8-s + (0.766 + 0.642i)9-s + (4.10 + 7.11i)10-s + (−0.287 + 0.498i)11-s + (2.67 − 0.974i)12-s + (4.60 − 3.86i)13-s + (1.09 + 1.89i)14-s + (0.647 − 3.67i)15-s + (−0.273 + 1.55i)16-s + (2.33 + 1.95i)17-s + ⋯
L(s)  = 1  + (−1.46 + 0.532i)2-s + (0.542 + 0.197i)3-s + (1.09 − 0.916i)4-s + (−0.289 − 1.64i)5-s − 0.899·6-s + (−0.0651 − 0.369i)7-s + (−0.331 + 0.573i)8-s + (0.255 + 0.214i)9-s + (1.29 + 2.24i)10-s + (−0.0867 + 0.150i)11-s + (0.773 − 0.281i)12-s + (1.27 − 1.07i)13-s + (0.292 + 0.506i)14-s + (0.167 − 0.948i)15-s + (−0.0683 + 0.387i)16-s + (0.566 + 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.871 + 0.490i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :1/2),\ 0.871 + 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577345 - 0.151249i\)
\(L(\frac12)\) \(\approx\) \(0.577345 - 0.151249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (-5.46 - 2.67i)T \)
good2 \( 1 + (2.06 - 0.753i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.647 + 3.67i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.172 + 0.977i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (0.287 - 0.498i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.60 + 3.86i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.33 - 1.95i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (4.33 + 1.57i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-2.34 - 4.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.56 + 4.43i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.37T + 31T^{2} \)
41 \( 1 + (4.49 - 3.77i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 6.49T + 43T^{2} \)
47 \( 1 + (-1.14 - 1.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.389 - 2.20i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (1.12 - 6.37i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (5.89 - 4.94i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.00 - 11.3i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-14.8 - 5.41i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 4.63T + 73T^{2} \)
79 \( 1 + (-0.535 - 3.03i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.57 + 4.68i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (0.0489 - 0.277i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.44 + 2.49i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.35133654444942535480865614212, −12.75591215014126827805589007690, −11.05672634820504608010266288540, −9.983199351515627827466244554447, −8.940413783069326922523229312026, −8.364528688176217686460734934754, −7.54094843436750366992849574729, −5.77701126042783729280402082605, −4.06595452535346644700556996447, −1.13688125828618045373386473821, 2.18188837720966597091955839553, 3.48578571739846228564118932571, 6.45092185741512904223932173721, 7.39294700987745717527317304522, 8.522257246287154798942067040338, 9.394175904586182156964170660228, 10.73391291706207192899013736587, 11.04473798855959024835646761329, 12.32922565176096046285508026099, 13.97951139631978524214918551144

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