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Properties

Label 2-110-11.3-c1-0-3
Degree $2$
Conductor $110$
Sign $-0.0320 + 0.999i$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
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  + (0.809 − 0.587i)2-s + (−0.650 − 2.00i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (−1.70 − 1.23i)6-s + (−0.675 + 2.07i)7-s + (−0.309 − 0.951i)8-s + (−1.15 + 0.841i)9-s − 10-s + (1.92 + 2.69i)11-s − 2.10·12-s + (5.11 − 3.71i)13-s + (0.675 + 2.07i)14-s + (−0.650 + 2.00i)15-s + (−0.809 − 0.587i)16-s + (−1.34 − 0.980i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.375 − 1.15i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (−0.695 − 0.505i)6-s + (−0.255 + 0.785i)7-s + (−0.109 − 0.336i)8-s + (−0.386 + 0.280i)9-s − 0.316·10-s + (0.581 + 0.813i)11-s − 0.607·12-s + (1.41 − 1.03i)13-s + (0.180 + 0.555i)14-s + (−0.167 + 0.516i)15-s + (−0.202 − 0.146i)16-s + (−0.327 − 0.237i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0320 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $-0.0320 + 0.999i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ -0.0320 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.832391 - 0.859550i\)
\(L(\frac12)\) \(\approx\) \(0.832391 - 0.859550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-1.92 - 2.69i)T \)
good3 \( 1 + (0.650 + 2.00i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.675 - 2.07i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-5.11 + 3.71i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.34 + 0.980i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.82 - 5.60i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.26T + 23T^{2} \)
29 \( 1 + (1.95 - 6.00i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.74 - 2.72i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.849 - 2.61i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.89 + 8.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.29T + 43T^{2} \)
47 \( 1 + (-0.582 - 1.79i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.36 - 4.62i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.47 + 7.63i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.51 - 2.55i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.61T + 67T^{2} \)
71 \( 1 + (11.1 + 8.10i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.32 + 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.40 - 1.02i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.25 + 5.27i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (9.72 - 7.06i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.03235290782806362180391455517, −12.39325572033259848767265856262, −11.82059917985809652901718436237, −10.54789210899685806361513179030, −9.056881361958071134853068963183, −7.72996963386780191064911582078, −6.44825516578789026062001732864, −5.49954960020109966081034279669, −3.63049455671947928067562183886, −1.60723876779165351558593010346, 3.68825457008564817390724446662, 4.31216959498749642487471087521, 5.92668513505927428335398998632, 7.00559574022296404104842225870, 8.580069537788415620838678924642, 9.754541321505697330857018621409, 11.22861070096016187705338665510, 11.32445193223612458914362496832, 13.25924323600472505776993372078, 13.92606969781827734011197713882

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