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Properties

Label 2-1110-111.68-c1-0-18
Degree $2$
Conductor $1110$
Sign $0.0872 + 0.996i$
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 + (−1.41 − i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (1.70 − 0.292i)6-s − 4·7-s + (0.707 + 0.707i)8-s + (1.00 + 2.82i)9-s − 1.00·10-s + 1.41·11-s + (−1.00 + 1.41i)12-s + (−3 + 3i)13-s + (2.82 − 2.82i)14-s + (−0.292 − 1.70i)15-s − 1.00·16-s + (1.41 + 1.41i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.816 − 0.577i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (0.696 − 0.119i)6-s − 1.51·7-s + (0.250 + 0.250i)8-s + (0.333 + 0.942i)9-s − 0.316·10-s + 0.426·11-s + (−0.288 + 0.408i)12-s + (−0.832 + 0.832i)13-s + (0.755 − 0.755i)14-s + (−0.0756 − 0.440i)15-s − 0.250·16-s + (0.342 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0872 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0872 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.0872 + 0.996i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.0872 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4282295428\)
\(L(\frac12)\) \(\approx\) \(0.4282295428\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (6 + i)T \)
good7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \)
19 \( 1 + (-4 + 4i)T - 19iT^{2} \)
23 \( 1 + (-1.41 - 1.41i)T + 23iT^{2} \)
29 \( 1 + (-2.82 + 2.82i)T - 29iT^{2} \)
31 \( 1 + (3 + 3i)T + 31iT^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 + 1.41iT - 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 + (8.48 + 8.48i)T + 59iT^{2} \)
61 \( 1 + (6 + 6i)T + 61iT^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + (-5 + 5i)T - 79iT^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + (1.41 - 1.41i)T - 89iT^{2} \)
97 \( 1 + (-6 + 6i)T - 97iT^{2} \)
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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

−9.611315291123783975646318418060, −9.051742017795322477606519952128, −7.65647354005068150731891413152, −6.97239273024662659244090820555, −6.49032805634132394371860172616, −5.72343930962737497177682735936, −4.73010517988797683512350664556, −3.23517271497079398645015930388, −1.89721627541678866850102460989, −0.29347935697828269337349601110, 1.07598828802837731205861724513, 2.95417104135067089236185922589, 3.63805441140259002068395332658, 4.93523821512521970069000322291, 5.75289344278902563946886756902, 6.66638591006437391324509327225, 7.47131193638748382339043718047, 8.817911709240217777892881106935, 9.465627601281878979723435432756, 10.13145838804810517787459047232

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