Properties

Label 2-1110-37.11-c1-0-10
Degree $2$
Conductor $1110$
Sign $0.806 - 0.590i$
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  + (0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·6-s + (−0.979 + 1.69i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 2.45·11-s + (0.499 + 0.866i)12-s + (1.23 + 0.710i)13-s + 1.95i·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (−0.831 + 0.479i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 0.408i·6-s + (−0.370 + 0.641i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 0.740·11-s + (0.144 + 0.249i)12-s + (0.341 + 0.197i)13-s + 0.523i·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (−0.201 + 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.806 - 0.590i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.806 - 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.230792547\)
\(L(\frac12)\) \(\approx\) \(2.230792547\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-5.73 + 2.01i)T \)
good7 \( 1 + (0.979 - 1.69i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 + (-1.23 - 0.710i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.831 - 0.479i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.95 - 2.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.15iT - 23T^{2} \)
29 \( 1 - 8.68iT - 29T^{2} \)
31 \( 1 - 6.99iT - 31T^{2} \)
41 \( 1 + (4.88 - 8.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 3.84iT - 43T^{2} \)
47 \( 1 - 13.5T + 47T^{2} \)
53 \( 1 + (5.33 + 9.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.41 + 3.70i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.96 + 4.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.23 - 7.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.35 - 2.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.05T + 73T^{2} \)
79 \( 1 + (-3.52 - 2.03i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.77 + 4.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.26 + 1.30i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.53iT - 97T^{2} \)
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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

−10.03137844447152948529315789066, −9.249577063582995171980797547607, −8.596410193693032778119044220610, −7.10472647883920891913325616897, −6.33153042595468975523150519605, −5.62068403899544145318881793191, −4.75174468075957001662552552115, −3.66730623957477434999628194171, −2.85434595192400078838598555308, −1.45452347008478091322773063670, 0.935980435007520130971070935246, 2.42519511659805499858699691931, 3.69855594324646518635501968961, 4.56118982919294951513526647457, 5.74401719869839856215226304605, 6.22338242854590563297964413134, 7.22721272334125949261083402558, 7.75169722019411018150354467033, 8.958355751894266589480521115087, 9.683684407415884930317707214544

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