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Properties

Label 2-1110-111.68-c1-0-14
Degree $2$
Conductor $1110$
Sign $-0.837 - 0.546i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + (−0.707 + 0.707i)2-s + (0.169 + 1.72i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−1.33 − 1.09i)6-s + 1.82·7-s + (0.707 + 0.707i)8-s + (−2.94 + 0.584i)9-s − 1.00·10-s + 5.15·11-s + (1.72 − 0.169i)12-s + (−3.16 + 3.16i)13-s + (−1.29 + 1.29i)14-s + (−1.09 + 1.33i)15-s − 1.00·16-s + (−2.98 − 2.98i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.0979 + 0.995i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (−0.546 − 0.448i)6-s + 0.690·7-s + (0.250 + 0.250i)8-s + (−0.980 + 0.194i)9-s − 0.316·10-s + 1.55·11-s + (0.497 − 0.0489i)12-s + (−0.878 + 0.878i)13-s + (−0.345 + 0.345i)14-s + (−0.283 + 0.345i)15-s − 0.250·16-s + (−0.724 − 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.837 - 0.546i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.342454427\)
\(L(\frac12)\) \(\approx\) \(1.342454427\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.169 - 1.72i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-0.159 - 6.08i)T \)
good7 \( 1 - 1.82T + 7T^{2} \)
11 \( 1 - 5.15T + 11T^{2} \)
13 \( 1 + (3.16 - 3.16i)T - 13iT^{2} \)
17 \( 1 + (2.98 + 2.98i)T + 17iT^{2} \)
19 \( 1 + (2.92 - 2.92i)T - 19iT^{2} \)
23 \( 1 + (-3.68 - 3.68i)T + 23iT^{2} \)
29 \( 1 + (-4.89 + 4.89i)T - 29iT^{2} \)
31 \( 1 + (-6.21 - 6.21i)T + 31iT^{2} \)
41 \( 1 + 0.816T + 41T^{2} \)
43 \( 1 + (2.69 - 2.69i)T - 43iT^{2} \)
47 \( 1 - 6.46iT - 47T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 + (0.207 + 0.207i)T + 59iT^{2} \)
61 \( 1 + (8.45 + 8.45i)T + 61iT^{2} \)
67 \( 1 - 12.0iT - 67T^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + 10.9iT - 73T^{2} \)
79 \( 1 + (-5.32 + 5.32i)T - 79iT^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 + (6.37 - 6.37i)T - 89iT^{2} \)
97 \( 1 + (8.32 - 8.32i)T - 97iT^{2} \)
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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

−9.934035410036007404625535332060, −9.321853920266664337581380024416, −8.775910654983400569548111763188, −7.82766862266770189868602897430, −6.71830722980200085394380326705, −6.16884726538893741728683752520, −4.77659635896162942145984859092, −4.45118943463429961382354801508, −2.97319001409912494650104030184, −1.60925667783646943483333358090, 0.72169457131654457950426644993, 1.82334678190894134847119858953, 2.72651149595209899080992547849, 4.14660025940369690917785968899, 5.19236008460313352138605198311, 6.48018535719134209363958358897, 6.95941204788821851968988136846, 8.154455206162572224905694306245, 8.597112046830022291827844152833, 9.326520290218264845045284767728

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