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Properties

Label 2-1110-37.27-c1-0-10
Degree $2$
Conductor $1110$
Sign $0.713 - 0.700i$
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 + (0.293 + 0.507i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 1.58·11-s + (0.499 − 0.866i)12-s + (3.63 − 2.09i)13-s + 0.586i·14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + (1.60 + 0.927i)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 + (0.110 + 0.191i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s − 0.478·11-s + (0.144 − 0.249i)12-s + (1.00 − 0.581i)13-s + 0.156i·14-s + (0.223 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.389 + 0.225i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.713 - 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.022410708\)
\(L(\frac12)\) \(\approx\) \(2.022410708\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.91 - 5.33i)T \)
good7 \( 1 + (-0.293 - 0.507i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
13 \( 1 + (-3.63 + 2.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.60 - 0.927i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.01 + 4.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.91iT - 23T^{2} \)
29 \( 1 - 3.75iT - 29T^{2} \)
31 \( 1 - 10.5iT - 31T^{2} \)
41 \( 1 + (-2.38 - 4.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.33iT - 43T^{2} \)
47 \( 1 - 5.85T + 47T^{2} \)
53 \( 1 + (-4.55 + 7.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.12 - 1.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.12 + 2.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.71 + 6.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.17 - 2.02i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + (5.59 - 3.23i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.161 + 0.279i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (12.9 + 7.49i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−10.09092994858002528986689008275, −8.864589618272450094632123213915, −8.085898572954958922388027309890, −7.31058516316629421554174199082, −6.67664911056524101183736904127, −5.45660901074562456339924596356, −5.18331190666886689904441089082, −3.61654151188787658076544354919, −2.94727433787011166942052678072, −1.27576510383218064158275219227, 0.910638515637390531897433343264, 2.53618605960847987110965481399, 3.80654900968377821397820600017, 4.27749648959251020853958367604, 5.45885942927382227788155915268, 6.00076776635298526743100328889, 7.22977047540087206643237342755, 8.061225175813985855920452430112, 9.100065280733388456535219327859, 9.896907950462779178855038386081

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