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Properties

Label 2-1110-185.142-c1-0-4
Degree $2$
Conductor $1110$
Sign $-0.885 + 0.463i$
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  + i·2-s + (0.707 + 0.707i)3-s − 4-s + (−2.07 + 0.825i)5-s + (−0.707 + 0.707i)6-s + (1.45 + 1.45i)7-s − i·8-s + 1.00i·9-s + (−0.825 − 2.07i)10-s + 3.60i·11-s + (−0.707 − 0.707i)12-s + 0.566i·13-s + (−1.45 + 1.45i)14-s + (−2.05 − 0.886i)15-s + 16-s − 3.56·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.929 + 0.368i)5-s + (−0.288 + 0.288i)6-s + (0.550 + 0.550i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.260 − 0.657i)10-s + 1.08i·11-s + (−0.204 − 0.204i)12-s + 0.157i·13-s + (−0.389 + 0.389i)14-s + (−0.530 − 0.228i)15-s + 0.250·16-s − 0.864·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.885 + 0.463i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.885 + 0.463i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9201073363\)
\(L(\frac12)\) \(\approx\) \(0.9201073363\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (2.07 - 0.825i)T \)
37 \( 1 + (3.08 - 5.24i)T \)
good7 \( 1 + (-1.45 - 1.45i)T + 7iT^{2} \)
11 \( 1 - 3.60iT - 11T^{2} \)
13 \( 1 - 0.566iT - 13T^{2} \)
17 \( 1 + 3.56T + 17T^{2} \)
19 \( 1 + (2.60 + 2.60i)T + 19iT^{2} \)
23 \( 1 - 2.32iT - 23T^{2} \)
29 \( 1 + (-0.784 + 0.784i)T - 29iT^{2} \)
31 \( 1 + (1.50 + 1.50i)T + 31iT^{2} \)
41 \( 1 - 8.53iT - 41T^{2} \)
43 \( 1 + 1.13iT - 43T^{2} \)
47 \( 1 + (4.15 + 4.15i)T + 47iT^{2} \)
53 \( 1 + (4.48 - 4.48i)T - 53iT^{2} \)
59 \( 1 + (5.54 + 5.54i)T + 59iT^{2} \)
61 \( 1 + (5.08 + 5.08i)T + 61iT^{2} \)
67 \( 1 + (1.64 - 1.64i)T - 67iT^{2} \)
71 \( 1 + 3.57T + 71T^{2} \)
73 \( 1 + (0.352 + 0.352i)T + 73iT^{2} \)
79 \( 1 + (-0.260 - 0.260i)T + 79iT^{2} \)
83 \( 1 + (-2.52 + 2.52i)T - 83iT^{2} \)
89 \( 1 + (5.23 - 5.23i)T - 89iT^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−10.17461652873847353198801215407, −9.283304906439752762282652037704, −8.554810266973055665445826330171, −7.87612562043672514779987276544, −7.09531408728874672296268338254, −6.30055886569888544882223575918, −4.84297922382022538210839095008, −4.53309902380679019354550231798, −3.34723771241737419845450725844, −2.07403773285159965773676782012, 0.38408913110648855887784437101, 1.66518296986504354084667363415, 3.04400520158046786481087510812, 3.93591417336856959291758703223, 4.68456524575923768164161087957, 5.92024054858723478250071818876, 7.10572738049101751617118453580, 7.940090510279608277997306577402, 8.579731089287094352261114390146, 9.122051500978070296890470501189

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